A384419 -id:A384419 - cp's OEIS frontend

A384912 The number of unordered factorizations of n into exponentially squarefree prime powers (A384419).

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 9, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 12 2025

First differs from A384913 at n = 64.

			a(4) = 2 since 4 has 2 factorizations: 2^1 * 2^1 and 2^2, with squarefree exponents 1 and 2.

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

Cf. A005117, A073576, A209061, A384419.

Similar sequences: A000688, A046951, A050361, A050377, A188581, A188585, A322885, A370256, A384913, A384914, A384915, A384916.

s[n_] := s[n] = If[n == 0, 1, Sum[Sum[d * Abs[MoebiusMu[d]], {d, Divisors[j]}] * s[n-j], {j, 1, n}] / n]; (* Jean-François Alcover at A073576 *)
f[p_, e_] := s[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = if(n < 1, 1, sum(j = 1, n, sumdiv(j, d, d*issquarefree(d)) * s(n-j))/n);
a(n) = vecprod(apply(s, factor(n)[, 2]));

Multiplicative with a(p^e) = A073576(e).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 2.1069024289184419840496..., where f(x) = (1-x) / Product_{k>=1} (1-x^A005117(k)).

A384420 The number of exponentially squarefree prime powers (not including 1) that divide n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 4, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 4, 1, 2, 3, 6, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 2, 3, 1, 4, 3, 2, 1, 4, 2, 2, 2

Offset: 1

Amiram Eldar, May 28 2025

The number of terms in A384419 that are larger than 1 and divide n.

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

Cf. A070321, A077761, A378085, A384419, A384421.

s[n_] := Module[{k = n}, While[! SquareFreeQ[k], k--]; k]; f[p_, e_] := s[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = while(!issquarefree(n), n--); n;
a(n) = vecsum(apply(s, factor(n)[, 2]));

Additive with a(p^e) = A070321(e).

Sum_{k=1..n} a(k) ~ n*(log(log(n)) + B + C), where B is Mertens's constant (A077761), and C = Sum_{p prime, e>=2} A378085(e-1)/p^e = 0.72770645470600638249... .

A384421 The number of exponentially squarefree prime powers (not including 1) that unitarily divide n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 0, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2, 3, 1, 1, 0, 2, 1, 3, 2, 2, 2

Offset: 1

Amiram Eldar, May 28 2025

First differs from A125029 at n = 64.
A number k unitarily divides n if k|n and gcd(k, n/k) = 1.
The number of unitary divisors of n that are larger than 1 and are terms in A384419.

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

Cf. A008966, A125029, A383959, A384419, A384420.

f[p_, e_] := If[SquareFreeQ[e], 1, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecsum(apply(issquarefree, factor(n)[, 2]));

Additive with a(p^e) = A008966(e).

Sum_{k=1..n} a(k) ~ n*(log(log(n)) + B - C + D), where B is Mertens's constant (A077761), C = Sum_{p prime} 1/p^2 (A085548), and D = Sum_{p prime, e>=2} (1-1/p)*A008966(e)/p^e = 0.40780808646244052181... .

cp's OEIS Frontend

A384912 The number of unordered factorizations of n into exponentially squarefree prime powers (A384419).

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A384420 The number of exponentially squarefree prime powers (not including 1) that divide n.

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A384421 The number of exponentially squarefree prime powers (not including 1) that unitarily divide n.

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