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 A242031 Numbers n such that prime factorization n = p_1^k_1*p_2^k_2*...*p_r^k_r satisfies k_1 >= k_2 >= ... >= k_r. 9
 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Complement sequence begins 18, 50, 54, 75, 90, 98, ... (A071365). Choie et al. call these "Hardy-Ramanujan integers". Hardy and Ramanujan show that the number of members of the sequence <= x is exp(2*Pi*sqrt(log(x)/(3*log log x))*(1+o(1))). - Robert Israel, Aug 18 2014 REFERENCES G. H. Hardy and S. Ramanujan, "Asymptotic Formulae for the Distribution of Integers of Various Types", Proc. London Math. Society, 2, XVI (1917), 112-132, in Collected Papers of Srinivasa Ramanujan, Chelsea, 1962, pages 245-261. LINKS Jens Kruse Andersen, Table of n, a(n) for n = 1..10000 Y. Choie, N. Lichiardopol, P. Moree, P. Sole, On Robin's criterion for the Riemann hypothesis J. Theor. Nombr. Bord. 19 (2) (2007), 357-372 Steven R. Finch, Errata and Addenda to Mathematical Constants. p. 9. Steven R. Finch, Errata and Addenda to Mathematical Constants, January 22, 2016. [Cached copy, with permission of the author] L. B. Richmond, Asymptotic results for partitions (I) and the distribution of certain integers. (1976) p. 388. EXAMPLE 12 = 2^2*3^1 is in the sequence, but 18 = 2^1*3^2 is not. MAPLE filter:= proc(n) local F; F:= ifactors(n)[2]; F:= sort(F, (s, t) -> s[1]>t[1]); ListTools:-Sorted(map(t -> t[2], F)); end: select(filter, [\$1..100]); # Robert Israel, Aug 18 2014 MATHEMATICA Select[Range[100], GreaterEqual @@ (FactorInteger[#][[All, 2]]) &] PROG (PARI) s=[]; for(n=1, 10^3, m=factor(n)[, 2]; if(vecsort(m, , 4)==m, s=concat(s, n))); s \\ Jens Kruse Andersen, Aug 18 2014 CROSSREFS Cf. A242061, A071365. Sequence in context: A151764 A093618 A317257 * A109427 A249830 A300473 Adjacent sequences:  A242028 A242029 A242030 * A242032 A242033 A242034 KEYWORD nonn,changed AUTHOR Jean-François Alcover, Aug 14 2014 STATUS approved

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Last modified April 24 10:11 EDT 2019. Contains 322420 sequences. (Running on oeis4.)