A369163 a(n) = A000005(A000688(n)).

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1

Offset

1,4

Comments

First differs from A007424, A278908, A307848, A323308, A358260 and A365549 at n = 36.

The sums of the first 10^k terms, for k = 1, 2, ..., are 13, 143, 1486, 15054, 151067, 1511982, 15123465, 151245456, 1512484372, 15124927227, ... . From these values the asymptotic mean of this sequence, whose existence was proven by Ivić (1983) (see the Formula section), can be empirically evaluated by 1.512... .

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. See p. 131, eq. 4.3.

Formula

Sum_{k=1..n} a(k) = c * n + O(sqrt(n) * log(n)^4), where c = Sum_{k>=1} d(k) * A000005(k) is a constant, d(k) is the asymptotic density of the set {m | A000688(m) = k} (e.g., d(1) = A059956, d(2) = A271971, d(3) appears in A048109) (Ivić, 1983).

Mathematica

Table[DivisorSigma[0, FiniteAbelianGroupCount[n]], {n, 1, 100}]

Prog

(PARI) a(n) = numdiv(vecprod(apply(numbpart, factor(n)[, 2])));

Crossrefs

Cf. A000005, A000688.

Cf. A048109, A059956, A271971.

Cf. A369162, A369164, A369165.

Cf. A007424, A278908, A307848, A323308, A358260, A365549.

Sequence in context: A291119 A366309 A007424 * A323308 A365549 A278908

Adjacent sequences: A369160 A369161 A369162 * A369164 A369165 A369166

Keyword

nonn,easy

Author

Amiram Eldar, Jan 15 2024

Status

approved