A369170 Numbers k such that A000005(k) <= A000688(k).

1, 16, 32, 64, 81, 128, 243, 256, 512, 576, 625, 729, 768, 1024, 1152, 1280, 1296, 1536, 1600, 1728, 1792, 2048, 2187, 2304, 2401, 2560, 2592, 2816, 2916, 3072, 3125, 3136, 3200, 3328, 3456, 3584, 3888, 4096, 4352, 4608, 4864, 5120, 5184, 5632, 5832, 5888, 6144

Offset: 1

Author

Amiram Eldar, Jan 15 2024

Keywords

• nonn
• easy

Comments

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

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.

Crossrefs

Cf. A000005, A000688.

Subsequences: A369168, A369169.

Programs

Mathematica

Select[Range[6000], DivisorSigma[0, #] <= FiniteAbelianGroupCount[#] &]

PARI

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

Formula

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