A369168 Numbers k such that A000005(k) = A000688(k).

1, 16, 81, 625, 1296, 2401, 10000, 14641, 23040, 28561, 32256, 38400, 38416, 50625, 50688, 59904, 75264, 78336, 83521, 87552, 89600, 105984, 125440, 130321, 133632, 140800, 142848, 166400, 170496, 185856, 188928, 194481, 198144, 216576, 217600, 234256, 243200

Offset

1,2

Comments

The asymptotic density of this sequence is 0 (Ivić, 1983).

If k is a term, then every number with the same prime signature (A124832) as k is a term. The least term of each prime signature is given in A369169.

References

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter II, page 73.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Aleksandar Ivić, On the number of abelian groups of a given order and on certain related multiplicative functions, Journal of Number Theory, Vol. 16, No. 1 (1983), pp. 119-137.

Formula

x * log(log(x))/log(x) << N(x) << x / log(x)^(1-eps) for every 0 < eps < 1, where N(x) is the number of terms not exceeding x (Ivić, 1983).

Mathematica

Select[Range[250000], DivisorSigma[0, #] == FiniteAbelianGroupCount[#] &]

Prog

(PARI) is(n) = {my(e = factor(n)[, 2]); vecprod(apply(x -> x+1, e)) == vecprod(apply(numbpart, e)); }

Crossrefs

Cf. A000005, A000688, A124832.

Subsequence of A369170.

A369169 is a subsequence.

Sequence in context: A108941 A377022 A153157 * A366307 A113849 A046453

Adjacent sequences: A369165 A369166 A369167 * A369169 A369170 A369171

Keyword

nonn

Author

Amiram Eldar, Jan 15 2024

Status

approved