

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
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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 77908696digit 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 nth 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), 448469. Errata
Keith Briggs, Abundant numbers and the Riemann Hypothesis, Experimental Math., Vol. 16 (2006), p. 251256.
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. 357372
J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (#6, 2002), 534543.
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 * A319431 A258581 A108501
Adjacent sequences: A073748 A073749 A073750 * A073752 A073753 A073754


KEYWORD

nonn


AUTHOR

T. D. Noe, Aug 07 2002


STATUS

approved



