

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
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OFFSET

1,2


COMMENTS

Complement sequence begins 18, 50, 54, 75, 90, 98, ... (A071365).
Choie et al. call these "HardyRamanujan 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), 112132, in Collected Papers of Srinivasa Ramanujan, Chelsea, 1962, pages 245261.


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), 357372
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

JeanFrançois Alcover, Aug 14 2014


STATUS

approved



