A369168 -id:A369168 - cp's OEIS frontend

A369169 Terms k of A025487 such that A000005(k) = A000688(k).

1, 16, 1296, 23040, 810000, 7257600, 16934400, 283852800, 1437004800, 1944810000, 13970880000, 30735936000, 232475443200, 852409958400, 1765360396800, 3269185920000, 7192209024000, 8029628006400, 28473963210000, 97893956160000, 181803061440000, 1086822696960000

Offset: 1

Amiram Eldar, Jan 15 2024

nonn

Since both A000005(k) and A000688(k) depend only on the prime signature of k (A124832), if k is a term of this sequence then every number m such that A046523(m) = k is a term of A369168.
From David A. Corneth, Jan 15 2024: (Start)
16 | a(n) for n > 1.
This sequence contains A002110(n)^4. (End)

			16 is in the sequence as 16 has 5 divisors (1, 2, 4, 8, 16) and 5 factorizations into prime powers (16 = 2*8 = 4*4 = 2*2*4 = 2*2*2*2).

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..377
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, A002110, A046523, A124832.

Intersection of A025487 and A369168.

lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]; Select[lps, DivisorSigma[0, #] == FiniteAbelianGroupCount[#] &]

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

A369169 Terms k of A025487 such that A000005(k) = A000688(k).

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