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

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

Offset: 1

Author

T. D. Noe, Aug 07 2002

Keywords

• nonn

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's 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), pp. 357-372.
• J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, arXiv:math/0008177 [math.NT], 2000-2001; Am. Math. Monthly 109 (#6, 2002), 534-543.
• T. Schwabhäuser, Preventing Exceptions to Robin's Inequality, arXiv preprint arXiv:1308.3678 [math.NT], 2013.
• Eric Weisstein's World of Mathematics, Colossally Abundant Number

Crossrefs

Cf. A004490.

Programs

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