A369165 -id:A369165 - cp's OEIS frontend

A369162 a(n) = A000688(A000688(n)).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jan 15 2024

First differs from A364388 at n = 42.

The sums of the first 10^k terms, for k = 1, 2, ..., are 10, 102, 1024, 10285, 102988, 1030280, 10304021, 103043644, 1030448091, 10304515936, ... . 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 1.0304... .

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter XIII, page 477-478.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Erdős and Aleksandar Ivić, On the iterates of the enumerating function of finite Abelian groups, Bull. Cl. Sci. Math. Nat., Sci. Math., Vol. 17 (1989), pp. 13-22; alternative link.
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. 4.2.

Cf. A000688, A364388.
Cf. A048109, A059956, A271971.
Cf. A369163, A369164, A369165, A369167.

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

A000688(n) = vecprod(apply(numbpart, factor(n)[, 2]));
a(n) = A000688(A000688(n));

Sum_{k=1..n} a(k) = c * n + O(sqrt(n) * log(n)^4), where c = Sum_{k>=1} d(k) * A000688(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).

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

Original entry on oeis.org

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

Amiram Eldar, Jan 15 2024

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... .

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. See p. 131, eq. 4.3.

Cf. A000005, A000688.
Cf. A048109, A059956, A271971.
Cf. A369162, A369164, A369165.
Cf. A007424, A278908, A307848, A323308, A358260, A365549.

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

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

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).

A369164 a(n) = A001221(A000688(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0

Offset: 1

Amiram Eldar, Jan 15 2024

First differs from A369165 at n = 36, from A080733 at n = 49, and from A107078 at n = 72.

The sums of the first 10^k terms, for k = 1, 2, ..., are 3, 40, 426, 4307, 43203, 432211, 4322486, 43226028, 432261887, 4322622387, ... . 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 0.43226... .

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter V, page 164.

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. 4.4.

Cf. A001221, A000688, A080733, A107078.
Cf. A048109, A059956, A271971.
Cf. A369162, A369163, A369165.

Table[PrimeNu[FiniteAbelianGroupCount[n]], {n, 1, 100}]

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

Sum_{k=1..n} a(k) = c * n + O(sqrt(n) * log(n)^3/log(log(n))^2), where c = Sum_{k>=1} d(k) * A001221(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).

cp's OEIS Frontend

A369162 a(n) = A000688(A000688(n)).

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

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A369164 a(n) = A001221(A000688(n)).

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