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A073751 Prime numbers that when multiplied in order yield the sequence of colossally abundant numbers A004490. 9
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, 5, 61, 67, 71, 73, 11, 79, 2, 83, 3, 89, 97, 13, 101, 103, 107, 109, 113, 127, 131, 137, 139, 2, 149, 151, 7, 157, 163, 167, 17, 173, 179, 181, 191, 193, 197, 199, 19, 211, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

This calculation assumes that the ratio of consecutive colossally abundant numbers is always prime, which is implied by a conjecture mentioned in Lagarias' paper.

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.

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.

From Robert G. Wilson v, May 30 2014: (Start)

First occurrence of the n-th prime: 1, 2, 4, 7, 9, 10, 14, 15, 16, 18, 19, 22, 23, 24, 26, 27, 28, 30, 31, 32, ..., .

Positions of 2: 1, 3, 5, 8, 11, 17, 25, 36, 51, 77, 114, 178, 282, 461, 759, 1286, 2200, 3812, 6664, ..., .

Positions of 3: 2, 6, 12, 21, 38, 68, 132, 271, 595, 1356, 3191, 7775, ..., . (End)

LINKS

T. D. Noe, Table of n, a(n) for n=1..10000

L. Alaoglu and P. Erdos, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469. Errata

Keith Briggs, Abundant numbers and the Riemann Hypothesis, Experimental Math., Vol. 16 (2006), p. 251-256.

Young Ju Choie; Nicolas Lichiardopol; Pieter Moree; Patrick Solé, On Robin's criterion for the Riemann hypothesis, Journal de théorie des nombres de Bordeaux, 19 no. 2 (2007), p. 357-372

J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (#6, 2002), 534-543.

T. Schwabhäuser, Preventing Exceptions to Robin's Inequality, arXiv preprint arXiv:1308.3678, 2013

Eric Weisstein's World of Mathematics, Colossally Abundant Number

MATHEMATICA

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

CROSSREFS

Cf. A004490.

Sequence in context: A100761 A027748 A000705 * A258581 A108501 A166226

Adjacent sequences:  A073748 A073749 A073750 * A073752 A073753 A073754

KEYWORD

nonn

AUTHOR

T. D. Noe, Aug 07 2002

STATUS

approved

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Last modified December 11 19:09 EST 2017. Contains 295919 sequences.