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A073751 Prime numbers that when multiplied in order yield the sequence of colossally abundant numbers A004490. 9


%S 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 5,61,67,71,73,11,79,2,83,3,89,97,13,101,103,107,109,113,127,131,137,

%U 139,2,149,151,7,157,163,167,17,173,179,181,191,193,197,199,19,211,3

%N Prime numbers that when multiplied in order yield the sequence of colossally abundant numbers A004490.

%C 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 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 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.

%C Alaoglu and Erdős´s paper proves that the quotient of two consecutive colossally abundant numbers is either a prime or the product of two distinct primes.

%H T. D. Noe, <a href="/A073751/b073751.txt">Table of n, a(n) for n=1..10000</a>

%H L. Alaoglu and P. Erdos, <a href="http://www.renyi.hu/~p_erdos/1944-03.pdf">On highly composite and similar numbers,</a> Trans. Amer. Math. Soc., 56 (1944), 448-469. <a href="http://upforthecount.com/math/errata.html">Errata</a>

%H Keith Briggs, <a href="http://projecteuclid.org/euclid.em/1175789744">Abundant numbers and the Riemann Hypothesis</a>, Experimental Math., Vol. 16 (2006), p. 251-256.

%H Young Ju Choie; Nicolas Lichiardopol; Pieter Moree; Patrick Solé, <a href="http://www.numdam.org/item?id=JTNB_2007__19_2_357_0">On Robin’s criterion for the Riemann hypothesis</a>, Journal de théorie des nombres de Bordeaux, 19 no. 2 (2007), p. 357-372

%H 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 T. Schwabhäuser, <a href="http://arxiv.org/abs/1308.3678">Preventing Exceptions to Robin's Inequality</a>, arXiv preprint arXiv:1308.3678, 2013

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ColossallyAbundantNumber.html">Colossally Abundant Number</a>

%t 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 Cf. A004490.

%K nonn,changed

%O 1,1

%A _T. D. Noe_, Aug 07 2002

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Last modified April 16 10:15 EDT 2014. Contains 240577 sequences.