Search: id:A073751
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%I A073751
%S A073751 2,3,2,5,2,3,7,2,11,13,2,3,5,17,19,23,2,29,31,7,3,37,41,43,2,47,53,59,
%T A073751 5,61,67,71,73,11,79,2,83,3,89,97,13,101,103,107,109,113,127,131,137,
%U A073751 139,2,149,151,7,157,163,167,17,173,179,181,191,193,197,199,19,211,3
%N A073751 Primes numbers that when multiplied in order yield the sequence of colossally 
               abundant numbers A004490.
%C A073751 The Mathematica program presents a very fast method of computing the 
               factors of colossally abundant numbers. The 100th number has a sigma[n]/
               n ratio of 10.5681.
%C A073751 This calculation assumes that the ratio of consecutive colossally abundant 
               numbers is always prime, which is implied by a conjecture mentioned 
               in Lagarias' paper.
%C A073751 The ratio of consecutive colossally abundant numbers is prime for at 
               least the first 10^7 terms. The 10^7-th term is a 77908696-digit 
               number which has a sigma(n)/n value of 33.849.
%H A073751 T. D. Noe, <a href="b073751.txt">Table of n, a(n) for n=1..10000</a>
%H A073751 J. C. Lagarias, <a href="http://arXiv.org/abs/math.NT/0008177">An elementary 
               problem equivalent to the Riemann hypothesis</a>, Am. Math. Monthly 
               109 (#6, 2002), 534-543.
%H A073751 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               ColossallyAbundantNumber.html">Colossally Abundant Number</a>
%t A073751 pFactor[f_List] := Module[{p=f[[1]], k=f[[2]]}, N[Log[(p^(k+2)-1)/(p^(k+1)-1)]/
               Log[p]]-1]; maxN=100; f={{2, 1}, {3, 0}}; primes=1; lst={2}; x=Table[pFactor[f[[i]]], 
               {i, primes+1}]; For[n=2, n<=maxN, n++, i=Position[x, Max[x]][[1, 
               1]]; AppendTo[lst, f[[i, 1]]]; f[[i, 2]]++; If[i>primes, primes++; 
               AppendTo[f, {Prime[i+1], 0}]; AppendTo[x, pFactor[f[[ -1]]]]]; x[[i]]=pFactor[f[[i]]]]; 
               lst
%Y A073751 Cf. A004490.
%Y A073751 Sequence in context: A100761 A027748 A000705 this_sequence A108501 A166226 
               A088167
%Y A073751 Adjacent sequences: A073748 A073749 A073750 this_sequence A073752 A073753 
               A073754
%K A073751 nonn
%O A073751 1,1
%A A073751 T. D. Noe (noe(AT)sspectra.com), Aug 07 2002

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