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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-Fran├žois Alcover, Aug 14 2014

STATUS

approved

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