A369163 -id:A369163 - 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).

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).

A369165 a(n) = A001222(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, 2, 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 A369164 at n = 36.
The sums of the first 10^k terms, for k = 1, 2, ..., are 3, 42, 450, 4592, 46185, 462402, 4625478, 46258861, 462599818, 4626029362, ... . 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 0.4626... .
First differs from A056170 at n=128, 256, 384, 512, 640.... - R. J. Mathar, Jan 18 2024

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. 4.5.

Cf. A001222, A000688.
Cf. A048109, A059956, A271971.
Cf. A369162, A369163, A369164.

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

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

Sum_{k=1..n} a(k) = c * n + O(sqrt(n) * log(n)^3/log(log(n))), where c = Sum_{k>=1} d(k) * A001222(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).

A380398 The number of unitary divisors of n that are perfect powers (A001597).

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

Offset: 1

Amiram Eldar, Jan 23 2025

First differs from A368978 at n = 32, from A007424 and A369163 at n = 36, from A278908, A307848, A358260 and A365549 at n = 64, and from A323308 at n = 72.
a(n) depends only on the prime signature of n (A118914).
The record values are 2^k, for k = 0, 1, 2, ..., and they are attained at A061742(k).
The sum of unitary divisors of n that are perfect powers is A380400(n).

			a(4) = 2 since 4 have 2 unitary divisors that are perfect powers, 1 and 4 = 2^2.
a(72) = 3 since 72 have 3 unitary divisors that are perfect powers, 1, 8 = 2^3, and 9 = 3^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001597, A005117, A008683, A061742, A077610, A091050, A118914, A323308, A380399, A380400.

Cf. A007424, A278908, A307848, A323308, A358260, A365549, A368978, A369163.

ppQ[n_] := n == 1 || GCD @@ FactorInteger[n][[;; , 2]] > 1; a[n_] := DivisorSum[n, 1 &, CoprimeQ[#, n/#] && ppQ[#] &]; Array[a, 100]

a(n) = sumdiv(n, d, gcd(d, n/d) == 1 && (d == 1 || ispower(d)));

a(n) = Sum_{d|n, gcd(d, n/d) == 1} [d in A001597], where [] is the Iverson bracket.
a(n) = A091050(n) - A380399(n).
a(n) = 1 if and only if n is squarefree (A005117).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 - Sum_{k>=2} mu(k)*(zeta(k)/zeta(k+1) - 1) = 1.49341326536904597349..., where mu is the Moebius function (A008683).

cp's OEIS Frontend

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

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

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A369165 a(n) = A001222(A000688(n)).

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A380398 The number of unitary divisors of n that are perfect powers (A001597).

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