Sharedresearch / OddPrimSets.sagewsOpen in CoCalc
Author: Nathan McNew
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def lcmfirstints(n): return lcm([1..n]) def LPF(n): if n==1: return 1 return factor(n)[-1][0] def LPP(n): if n==1: return 0 f=factor(n) return max([p[0]^p[1] for p in f]) def CountPrimSets(V): #Returns the number of primitive subsets of the list of integers V if len(V)==0: return 1 components=ListConnectedComponents(V) count=1 for c in components: if len(c)==1: count*=2 elif len(c)==2: count*=3 elif len(c)==3: if not c[1]%c[0]==0 or not c[2]%c[1]==0: count*=5 else: count*=4 else: count*=sum(reduceCountPrimSubsets(c)) return count def ListConnectedComponents(V): #Break the list of integers V into connected components of the divisibility poset a=V[0] n=V[-1] L=dict() Components=[] for i in V: if i in L: #i has been visited, so its multiples #have already been taken into account continue L[i]=[i] Components.append(L[i]) if i>n/2: continue for j in range(2,int(floor(n/i))+1): h=i*j if not h in V: continue if not h in L: L[i].append(i*j) L[h]=L[i] continue elif L[i]==L[h]: continue L[h].extend(L[i]) Components.remove(L[i]) for k in reversed(L[i]): #Reversed so that i is processed last... L[k]=L[h] if len(Components)==1: return [V[:]] for c in Components: c.sort() return Components def reduceCountPrimSubsets(L): #Returns a pair, the count of primitive sets containing the first element of L, and the number not containing the first element of L V=L[:] if len(V)==0: return (0,1) a=V[0] n=V[-1] del V[0] countnothasa=CountPrimSets(V) if a==1: counthasa=1 else: for i in range(2,int(floor(n/a))+1): if (a*i) in V: V.remove(a*i) counthasa=CountPrimSets(V) return (counthasa,countnothasa) def seqgcd(L): g=L[0] for i in L: if not i%g==0: g=gcd(Integer(g),Integer(i)) if g==1: return 1 return g def is_pow(p,n): if (n^(1/p)).is_integer(): return true return false def largestPwrDiv(p,n): prod=1 for f in factor(n): prod*=f[0]**floor(f[1]/p) return prod def validDs(n): P=[] for p in Primes(): if p<n: P.append(p) else: break d=1 L=[1..n] grow=true while grow: grow=false for p in P: for l in L: if not p*l in L and p*l<d*(n+1): L.append(p*l) grow=true break #print L for p in P: for l in L: if not p.divides(l) and l/p > d: d*=p L=[k*p for k in L] L.append(l) grow=true break #print L #print d, return d def validOddDs(n): #Returns an integer, supported on the primes p<=n which is the largest at which new structure is added to the graph. #WARNING: this is only finite for small values of n. When n is sufficiently large this will run forever... P=[] for p in Primes(): if p<n: if not p==2: P.append(p) else: break d=1 L=[1..n] grow=true while grow: grow=false for p in P: for l in L: if not p*l in L and p*l<d*(n+1): L.append(p*l) grow=true break #print L for p in P: for l in L: if not p.divides(l) and l/p > d: d*=p L=[k*p for k in L] L.append(l) grow=true break #print L #print d, return d
[validOddDs(n) for n in [1..10]]
[1, 1, 1, 3, 3, 45, 45, 39375, 4921875, 1688203125]
newprodlog=0 accountedfor=0 # Total size of the subset of [0,1] that has been taken into account R=RealField(300) for n in range(1,11): #process the interval (1/(n+1),1/n] D=validOddDs(n) # For small values of n the potential values of d is a finite set, this returns the largest value of d that needs to be considered dlist=[d for d in divisors(D)] for d in dlist: valstoprocess=[[d,i] for i in range((d*n),(d*(n+1))) if i==(d*n) or LPF(i)<=n] vals=[i for i in range((d*n)+1,(d*(n+1)+1)) if i==(d*(n+1)) or LPF(i)<=n] prev=d*n L=[] for p in valstoprocess: v=p[1] if len(L)==0: I=[i for i in range(d,d*n+1) if not 2.divides(i)] L=sorted(ListConnectedComponents(I))[0] (counthasd,countnothasd)=reduceCountPrimSubsets(L[:]) else: vDivs=[j for j in divisors(v) if j>d and not j in L] if len(vDivs)>1: CurList=sorted(ListConnectedComponents([i for i in [d..v] if i%2==1]))[0] NewElements=[i for i in CurList if not i in L and not i==v] CountPrimSubsNewElements=sum(reduceCountPrimSubsets(NewElements)) counthasd*=CountPrimSubsNewElements countnothasd*=CountPrimSubsNewElements else: CurList=L+vDivs CurList.sort() (x,y)=reduceCountPrimSubsets([i for i in CurList if not Integer(i).divides(v)]) #Count sets which contain v counthasd+=x countnothasd+=y L=CurList nextv=vals.pop(0) r=1+counthasd/countnothasd print '\t',d,prev,r,N(r),prev,nextv, for pr in r.support(): newprodlog+=(d*euler_phi(D/d)*(nextv-prev)/(D*(prev)*(nextv)))*log(R(pr))*r.valuation(pr) print RR(newprodlog) accountedfor+=(d*euler_phi(D/d)*(nextv-prev)/(D*(prev)*(nextv))) prev=nextv print (n+1),d,d*n,RR(exp(newprodlog/2)), RR(exp(newprodlog/2)*sqrt(2)^(1.0-accountedfor)) print " "
1 1 2 2.00000000000000 1 2 0.346573590279973 2 1 1 1.18920711500272 1.41421356237309 1 2 2 2.00000000000000 2 3 0.462098120373297 3 1 2 1.25992104989487 1.41421356237310 1 3 3/2 1.50000000000000 3 4 0.495886879382311 4 1 3 1.28138745447418 1.39736292764435 1 4 3/2 1.50000000000000 4 5 0.509402382985916 5 1 4 1.29007607756724 1.39067902638730 3 12 3/2 1.50000000000000 12 15 0.516160134787719 5 3 12 1.29444245710919 1.38734907430991 1 5 5/4 1.25000000000000 5 6 0.521118880372479 6 1 5 1.29765584446511 1.38012285822089 3 15 4/3 1.33333333333333 15 16 0.522317555674361 3 16 4/3 1.33333333333333 16 18 0.524315347844165 6 3 15 1.29973146003083 1.37701751448039 1 6 5/4 1.25000000000000 6 7 0.527148916749743 7 1 6 1.30157420443377 1.37291441280781 3 18 4/3 1.33333333333333 18 20 0.528001308075526 3 20 13/10 1.30000000000000 20 21 0.528334469046278 7 3 18 1.30234597529844 1.37171473272409 5 30 5/4 1.25000000000000 30 32 0.528644390645326 5 32 5/4 1.25000000000000 32 35 0.529042861272672 7 5 30 1.30280734288341 1.37069176371900 9 54 4/3 1.33333333333333 54 60 0.529469056935564 9 60 13/10 1.30000000000000 60 63 0.529635637420940 7 9 54 1.30319353667153 1.37009276389166 15 90 4/3 1.33333333333333 90 96 0.529768823565594 15 96 4/3 1.33333333333333 96 100 0.529848735252386 15 100 21/16 1.31250000000000 100 105 0.529935063416032 7 15 90 1.30338865628789 1.36979544281102 45 270 4/3 1.33333333333333 270 288 0.530001656488358 45 288 4/3 1.33333333333333 288 300 0.530041612331754 45 300 21/16 1.31250000000000 300 315 0.530084776413577 7 45 270 1.30348622705110 1.36964680646698 1 7 9/8 1.12500000000000 7 8 0.531206519610305 8 1 7 1.30421752051641 1.36589934411652 3 21 11/9 1.22222222222222 21 24 0.531843569437169 8 3 21 1.30463301245798 1.36483202592733 5 35 23/20 1.15000000000000 35 36 0.531917517554828 5 36 23/20 1.15000000000000 36 40 0.532176335966633 8 5 35 1.30485009961720 1.36393318168343 9 63 11/9 1.22222222222222 63 64 0.532216151580812 9 64 11/9 1.22222222222222 64 70 0.532431155897379 9 70 11/9 1.22222222222222 70 72 0.532494860880065 8 9 63 1.30505792979905 1.36340018663033 15 105 5/4 1.25000000000000 105 108 0.532534216003754 15 108 5/4 1.25000000000000 108 112 0.532583409908365 15 112 5/4 1.25000000000000 112 120 0.532671958936664 8 15 105 1.30517349652719 1.36314592342332 45 315 5/4 1.25000000000000 315 320 0.532683027565201 45 320 5/4 1.25000000000000 320 324 0.532691636498508 45 324 5/4 1.25000000000000 324 336 0.532716233450814 45 336 5/4 1.25000000000000 336 343 0.532729786873512 45 343 5/4 1.25000000000000 343 350 0.532742798159303 45 350 5/4 1.25000000000000 350 360 0.532760507964963 8 45 315 1.30523128372887 1.36301880960214 1 8 9/8 1.12500000000000 8 9 0.533508336762782 9 1 8 1.30571941975462 1.36053145605606 3 24 11/9 1.22222222222222 24 25 0.533661228721229 3 25 6/5 1.20000000000000 25 27 0.533908182787574 9 3 24 1.30598048921039 1.35980571626954 5 40 23/20 1.15000000000000 40 42 0.533984243708595 5 42 8/7 1.14285714285714 42 45 0.534081137236123 9 5 40 1.30609343166146 1.35932495191523 7 56 9/8 1.12500000000000 56 60 0.534155920115904 7 60 9/8 1.12500000000000 60 63 0.534205775369092 9 7 56 1.30617482872113 1.35891120175435 9 72 11/9 1.22222222222222 72 75 0.534282221348316 9 75 29/24 1.20833333333333 75 80 0.534390359633824 9 80 29/24 1.20833333333333 80 81 0.534410385242251 9 9 72 1.30630846368978 1.35855189932410 15 120 5/4 1.25000000000000 120 125 0.534444388069118 15 125 49/40 1.22500000000000 125 126 0.534450278415610 15 126 67/55 1.21818181818182 126 128 0.534461466592150 15 128 116/95 1.22105263157895 128 135 0.534498450536503 9 15 120 1.30636598517581 1.35841243052336 21 168 11/9 1.22222222222222 168 175 0.534523932529578 21 175 83/69 1.20289855072464 175 180 0.534539571395457 21 180 83/69 1.20289855072464 180 189 0.534565636171922 9 21 168 1.30640987042732 1.35829200531545 25 200 23/20 1.15000000000000 200 210 0.534580848356126 25 210 8/7 1.14285714285714 210 216 0.534588922816754 25 216 55/48 1.14583333333333 216 224 0.534599212474677 25 224 39/34 1.14705882352941 224 225 0.534600456929294 9 25 200 1.30643261571588 1.35819610199706 35 280 23/20 1.15000000000000 280 288 0.534607851741060 35 288 23/20 1.15000000000000 288 294 0.534613133749464 35 294 55/48 1.14583333333333 294 300 0.534618072785267 35 300 39/34 1.14705882352941 300 315 0.534629687695025 9 35 280 1.30645170986828 1.35811633269476 45 360 5/4 1.25000000000000 360 375 0.534646689108458 45 375 79/64 1.23437500000000 375 378 0.534649744923702 45 378 27/22 1.22727272727273 378 384 0.534655549753992 45 384 187/152 1.23026315789474 384 392 0.534663101798307 45 392 133/108 1.23148148148148 392 400 0.534670386389599 45 400 69/56 1.23214285714286 400 405 0.534674804480898 9 45 360 1.30648118165172 1.35804735501843 63 504 11/9 1.22222222222222 504 512 0.534679781432670 63 512 11/9 1.22222222222222 512 525 0.534687545477435 63 525 197/162 1.21604938271605 525 540 0.534695825155467 63 540 17/14 1.21428571428571 540 560 0.534706097960464 63 560 415/342 1.21345029239766 560 567 0.534709510090679 9 63 504 1.30650385296147 1.35798791284141 75 600 5/4 1.25000000000000 600 625 0.534716310656052 75 625 49/40 1.22500000000000 625 630 0.534717488725351 75 630 67/55 1.21818181818182 630 640 0.534719726360658 75 640 116/95 1.22105263157895 640 648 0.534721487500866 75 648 11/9 1.22222222222222 648 672 0.534726543451873 75 672 214/175 1.22285714285714 672 675 0.534727151736199 9 75 600 1.30651537745122 1.35796004892577 105 840 5/4 1.25000000000000 840 864 0.534731087248568 105 864 5/4 1.25000000000000 864 875 0.534732818874011 105 875 47/38 1.23684210526316 875 882 0.534733847137978 105 882 459/373 1.23056300268097 882 896 0.534735807377872 105 896 685/557 1.22980251346499 896 900 0.534736354609614 105 900 911/741 1.22941970310391 900 945 0.534742182962967 9 105 840 1.30652519675257 1.35793705339430 125 1000 23/20 1.15000000000000 1000 1008 0.534742690035774 125 1008 23/20 1.15000000000000 1008 1024 0.534743680412350 125 1024 23/20 1.15000000000000 1024 1029 0.534743983588853 125 1029 23/20 1.15000000000000 1029 1050 0.534745225399808 125 1050 101/88 1.14772727272727 1050 1080 0.534746891718204 125 1080 31/27 1.14814814814815 1080 1120 0.534748980158695 125 1120 147/128 1.14843750000000 1120 1125 0.534749231228669 9 125 1000 1.30652980112905 1.35791793431147 175 1400 23/20 1.15000000000000 1400 1440 0.534750710191022 175 1440 23/20 1.15000000000000 1440 1458 0.534751349248829 175 1458 23/20 1.15000000000000 1458 1470 0.534751766592703 175 1470 39/34 1.14705882352941 1470 1500 0.534752762156396 175 1500 101/88 1.14772727272727 1500 1512 0.534753150964022 175 1512 31/27 1.14814814814815 1512 1536 0.534753912374617 175 1536 147/128 1.14843750000000 1536 1568 0.534754893116701 175 1568 85/74 1.14864864864865 1568 1575 0.534755102619594 9 175 1400 1.30653363670829 1.35790200047895 225 1800 5/4 1.25000000000000 1800 1875 0.534758502902280 225 1875 79/64 1.23437500000000 1875 1890 0.534759114065329 225 1890 27/22 1.22727272727273 1890 1920 0.534760275031387 225 1920 187/152 1.23026315789474 1920 1944 0.534761188735514 225 1944 133/108 1.23148148148148 1944 1960 0.534761788290353 225 1960 69/56 1.23214285714286 1960 2000 0.534763248965435 225 2000 53/43 1.23255813953488 2000 2016 0.534763817922708 225 2016 503/408 1.23284313725490 2016 2025 0.534764134359588 9 225 1800 1.30653953685766 1.35788821253207 315 2520 5/4 1.25000000000000 2520 2560 0.534765241222442 315 2560 5/4 1.25000000000000 2560 2592 0.534766102115772 315 2592 5/4 1.25000000000000 2592 2625 0.534766967928493 315 2625 259/208 1.24519230769231 2625 2646 0.534767498335101 315 2646 443/357 1.24089635854342 2646 2688 0.534768517959007 315 2688 1183/954 1.24004192872117 2688 2700 0.534768802542595 315 2700 740/597 1.23953098827471 2700 2744 0.534769822762355 315 2744 1777/1434 1.23919107391911 2744 2800 0.534771073251026 315 2800 1037/837 1.23894862604540 2800 2835 0.534771829029289 9 315 2520 1.30654456356243 1.35787683689410 375 3000 5/4 1.25000000000000 3000 3024 0.534772098892995 375 3024 5/4 1.25000000000000 3024 3072 0.534772625970544 375 3072 5/4 1.25000000000000 3072 3087 0.534772787320814 375 3087 5/4 1.25000000000000 3087 3125 0.534773189142364 375 3125 139/112 1.24107142857143 3125 3136 0.534773299963387 375 3136 47/38 1.23684210526316 3136 3150 0.534773437677311 375 3150 206/167 1.23353293413174 3150 3200 0.534773913601022 375 3200 85/69 1.23188405797101 3200 3240 0.534774281404817 375 3240 16/13 1.23076923076923 3240 3360 0.534775327710510 375 3360 353/287 1.22996515679443 3360 3375 0.534775452872016 9 375 3000 1.30654693092058 1.35787132936867 525 4200 5/4 1.25000000000000 4200 4320 0.534776239974490 525 4320 5/4 1.25000000000000 4320 4374 0.534776580080497 525 4374 5/4 1.25000000000000 4374 4375 0.534776586299579 525 4375 263/212 1.24056603773585 4375 4410 0.534776794860977 525 4410 21/17 1.23529411764706 4410 4480 0.534777194160096 525 4480 835/677 1.23338257016248 4480 4500 0.534777305144462 525 4500 1061/861 1.23228803716609 4500 4536 0.534777501614899 525 4536 117/95 1.23157894736842 4536 4608 0.534777884288774 525 4608 1513/1229 1.23108218063466 4608 4704 0.534778375345095 525 4704 1739/1413 1.23071479122435 4704 4725 0.534778479953375 9 525 4200 1.30654890843401 1.35786674467796 625 5000 23/20 1.15000000000000 5000 5040 0.534778606721577 625 5040 23/20 1.15000000000000 5040 5103 0.534778802351518 625 5103 23/20 1.15000000000000 5103 5120 0.534778854315721 625 5120 23/20 1.15000000000000 5120 5145 0.534778930109846 625 5145 23/20 1.15000000000000 5145 5184 0.534779046888944 625 5184 23/20 1.15000000000000 5184 5250 0.534779240562585 625 5250 147/128 1.14843750000000 5250 5292 0.534779360119715 625 5292 85/74 1.14864864864865 5292 5376 0.534779593939908 625 5376 193/168 1.14880952380952 5376 5400 0.534779659475721 625 5400 54/47 1.14893617021277 5400 5488 0.534779895056713 625 5488 239/208 1.14903846153846 5488 5600 0.534780184364235 625 5600 131/114 1.14912280701754 5600 5625 0.534780247402273 9 625 5000 1.30655006306373 1.35786196878382 875 7000 23/20 1.15000000000000 7000 7056 0.534780331914408 875 7056 23/20 1.15000000000000 7056 7168 0.534780496977170 875 7168 23/20 1.15000000000000 7168 7200 0.534780543194744 875 7200 23/20 1.15000000000000 7200 7203 0.534780547506588 875 7203 23/20 1.15000000000000 7203 7290 0.534780671006305 875 7290 23/20 1.15000000000000 7290 7350 0.534780754475080 875 7350 147/128 1.14843750000000 7350 7500 0.534780955331059 875 7500 85/74 1.14864864864865 7500 7560 0.534781033545472 875 7560 193/168 1.14880952380952 7560 7680 0.534781186462370 875 7680 54/47 1.14893617021277 7680 7776 0.534781305492243 875 7776 239/208 1.14903846153846 7776 7840 0.534781383275885 875 7840 131/114 1.14912280701754 7840 7875 0.534781425301244 9 875 7000 1.30655083255594 1.35785878459196 1125 9000 5/4 1.25000000000000 9000 9072 0.534781560233096 1125 9072 5/4 1.25000000000000 9072 9216 0.534781823771871 1125 9216 5/4 1.25000000000000 9216 9261 0.534781904447006 1125 9261 5/4 1.25000000000000 9261 9375 0.534782105357781 1125 9375 259/208 1.24519230769231 9375 9408 0.534782161618767 1125 9408 169/136 1.24264705882353 9408 9450 0.534782231992538 1125 9450 367/296 1.23986486486486 9450 9600 0.534782475759424 1125 9600 223/180 1.23888888888889 9600 9604 0.534782482132213 1125 9604 525/424 1.23820754716981 9604 9720 0.534782664192727 1125 9720 151/122 1.23770491803279 9720 9800 0.534782787006703 1125 9800 683/552 1.23731884057971 9800 10000 0.534783085008074 1125 10000 381/308 1.23701298701299 10000 10080 0.534783200762954 1125 10080 841/680 1.23676470588235 10080 10125 0.534783265010531 9 1125 9000 1.30655203439335 1.35785604972906 1575 12600 5/4 1.25000000000000 12600 12800 0.534783486383102 1575 12800 5/4 1.25000000000000 12800 12960 0.534783658561768 1575 12960 5/4 1.25000000000000 12960 13122 0.534783828614772 1575 13122 5/4 1.25000000000000 13122 13125 0.534783831724313 1575 13125 339/272 1.24632352941176 13125 13230 0.534783938244907 1575 13230 563/453 1.24282560706402 13230 13440 0.534784143637483 1575 13440 1423/1146 1.24171029668412 13440 13500 0.534784200909891 1575 13500 860/693 1.24098124098124 13500 13608 0.534784302451285 1575 13608 2017/1626 1.24046740467405 13608 13720 0.534784405866202 1575 13720 1157/933 1.24008574490890 13720 13824 0.534784500258912 1575 13824 2611/2106 1.23979107312441 13824 14000 0.534784656632698 1575 14000 1454/1173 1.23955669224211 14000 14112 0.534784754026716 1575 14112 3205/2586 1.23936581593194 14112 14175 0.534784808095703 9 1575 12600 1.30655304245427 1.35785377746611 1875 15000 5/4 1.25000000000000 15000 15120 0.534784875561629 1875 15120 5/4 1.25000000000000 15120 15309 0.534784979675713 1875 15309 5/4 1.25000000000000 15309 15360 0.534785007331016 1875 15360 5/4 1.25000000000000 15360 15435 0.534785047668584 1875 15435 5/4 1.25000000000000 15435 15552 0.534785109818318 1875 15552 5/4 1.25000000000000 15552 15625 0.534785148123971 1875 15625 199/160 1.24375000000000 15625 15680 0.534785176105792 1875 15680 31/25 1.24000000000000 15680 15750 0.534785210947285 1875 15750 266/215 1.23720930232558 15750 15876 0.534785272238947 1875 15876 21/17 1.23529411764706 15876 16000 0.534785331183103 1875 16000 364/295 1.23389830508475 16000 16128 0.534785390757511 1875 16128 413/335 1.23283582089552 16128 16200 0.534785423718751 1875 16200 154/125 1.23200000000000 16200 16384 0.534785506368213 1875 16384 511/415 1.23132530120482 16384 16464 0.534785541633717 1875 16464 16/13 1.23076923076923 16464 16800 0.534785685767664 1875 16800 203/165 1.23030303030303 16800 16807 0.534785688703806 1875 16807 658/535 1.22990654205607 16807 16875 0.534785717055403 9 1875 15000 1.30655363625644 1.35785240264215 2625 21000 5/4 1.25000000000000 21000 21168 0.534785762032687 2625 21168 5/4 1.25000000000000 21168 21504 0.534785849878945 2625 21504 5/4 1.25000000000000 21504 21600 0.534785874475897 2625 21600 5/4 1.25000000000000 21600 21609 0.534785876770657 2625 21609 5/4 1.25000000000000 21609 21870 0.534785942497099 2625 21870 5/4 1.25000000000000 21870 21875 0.534785943740915 2625 21875 61/49 1.24489795918367 21875 21952 0.534785962474359 2625 21952 601/484 1.24173553719008 21952 22050 0.534785985853047 2625 22050 1285/1037 1.23915139826422 22050 22400 0.534786066891138 2625 22400 1511/1221 1.23751023751024 22400 22500 0.534786089441560 2625 22500 1737/1405 1.23629893238434 22500 22680 0.534786129346968 2625 22680 1963/1589 1.23536815607300 22680 23040 0.534786207010485 2625 23040 2189/1773 1.23463056965595 23040 23328 0.534786267244765 2625 23328 2415/1957 1.23403168114461 23328 23520 0.534786306490885 2625 23520 2641/2141 1.23353573096684 23520 23625 0.534786327643278 9 2625 21000 1.30655403513940 1.35785148922669 4375 35000 23/20 1.15000000000000 35000 35280 0.534786348771312 4375 35280 23/20 1.15000000000000 35280 35721 0.534786381376302 4375 35721 23/20 1.15000000000000 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Error in lines 4-42 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1188, in execute flags=compile_flags), namespace, locals) File "", line 18, in <module> File "", line 15, in ListConnectedComponents File "src/cysignals/signals.pyx", line 320, in cysignals.signals.python_check_interrupt KeyboardInterrupt