A369169 -id:A369169 - cp's OEIS frontend

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

Amiram Eldar, Jan 15 2024

nonn

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.

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

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.

Cf. A000005, A000688, A124832.
Subsequence of A369170.
A369169 is a subsequence.

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

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

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).

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

Original entry on oeis.org

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

Amiram Eldar, Jan 15 2024

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

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

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.

Cf. A000005, A000688.

Subsequences: A369168, A369169.

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

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

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

cp's OEIS Frontend

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

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