A370256 -id:A370256 - 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)).

A384913 The number of unordered factorizations of n into exponentially Fibonacci powers of primes (A115975).

Original entry on oeis.org

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, 8, 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 A384912 at n = 64.

			a(4) = 2 since 4 has 2 factorizations: 2^1 * 2^1 and 2^2, with exponents 1 and 2 that are Fibonacci numbers.

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

Cf. A000045, A003107, A115063, A115975.

Similar sequences: A000688, A046951, A050361, A050377, A188581, A188585, A322885, A370256, A384912, A384914, A384915, A384916.

fib[n_] := Boole[Or @@ IntegerQ /@ Sqrt[5*n^2 + {-4, 4}]];
s[n_] := s[n] = If[n == 0, 1, Sum[Sum[d * fib[d], {d, Divisors[j]}] * s[n-j], {j, 1, n}] / n];
f[p_, e_] := s[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

isfib(n) = issquare(5*n^2 - 4) || issquare(5*n^2 + 4);
s(n) = if(n < 1, 1, sum(j = 1, n, sumdiv(j, d, d*isfib(d)) * s(n-j))/n);
a(n) = vecprod(apply(s, factor(n)[, 2]));

Multiplicative with a(p^e) = A003107(e).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 2.05893526314055968638..., where f(x) = (1-x) / Product_{k>=2} (1-x^A000045(k)).

A384914 The number of unordered factorizations of n into numbers of the form p^(k^2) where p is prime and k >= 0 (A323520).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 12 2025

First differs from A203640, A295658 and A365333 at n = 64, from A043289 and A053164 at n = 81, and from A063775 at n = 512.

			a(16) = 2 since 4 has 2 factorizations: 2^1 * 2^1 * 2^1 * 2^1 and 2^4, with exponents 1 and 4 that are squares.

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

Cf. A000290, A001156, A197680, A323520.
Cf. A043289, A053164, A063775, A203640, A295658, A365333.
Similar sequences: A000688, A046951, A050361, A050377, A188581, A188585, A322885, A370256, A384912, A384913, A384915, A384916.

s[n_] := s[n] = If[n == 0, 1, Sum[Sum[d * Boole[IntegerQ[Sqrt[d]]], {d, Divisors[j]}] * s[n-j], {j, 1, n}] / n];
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*issquare(d)) * s(n-j))/n);
a(n) = vecprod(apply(s, factor(n)[, 2]));

Multiplicative with a(p^e) = A001156(e).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.08451356983124311685..., where f(x) = (1-x) / Product_{k>=1} (1-x^(k^2)).

A384915 The number of unordered factorizations of n into powers of primes of the form p^e where p is prime and 0 <= e <= p (A074583).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 12 2025

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

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

Cf. A026820, A048103, A074583.

Similar sequences: A000688, A046951, A050361, A050377, A188581, A188585, A322885, A370256, A384912, A384913, A384914, A384916.

f[p_, e_] := Length[IntegerPartitions[e, p]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

T(n, k)=my(s); forpart(v=n, s++, , k); s \\ Charles R Greathouse IV at A026820
a(n) = {my(f = factor(n)); prod(i = 1, #f~, T(f[i,2], f[i,1]));}

Multiplicative with a(p^e) = A026820(e, p).

a(n) >= A384916(n), with equality if and only if n is in A048103.

A384916 The number of unordered factorizations of n into powers of primes of the form p^e where p is prime and 0 <= e < p.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 12 2025

First differs from A298735 at n = 125.

			a(9) = 2 since 9 has 2 factorizations: 3^1 * 3^1 and 3^2, with exponents 1 and 2 that are < 3.

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

Cf. A026820, A048103, A298735.

Similar sequences: A000688, A046951, A050361, A050377, A188581, A188585, A322885, A370256, A384912, A384913, A384914, A384915.

f[p_, e_] := Length[IntegerPartitions[e, p-1]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

T(n, k)=my(s); forpart(v=n, s++, , k); s \\ Charles R Greathouse IV at A026820
a(n) = {my(f = factor(n)); prod(i = 1, #f~, T(f[i,2], f[i,1]-1));}

Multiplicative with a(p^e) = A026820(e, p-1).

a(n) <= A384915(n), with equality if and only if n is in A048103.

A057523 Number of ways n-th powerful (1) number can be expressed as b^2*c^3.

Original entry on oeis.org

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

Offset: 1

Henry Bottomley, Sep 02 2000

Always positive. (Would be zero for any non-powerful number.)

The positive terms of A370256. - Amiram Eldar, Feb 23 2024

			a(40) = 2 since the 40th powerful number is 576 = 24^2*1^3 = 3^2*4^3.

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

Cf. A001694, A002117, A370256.

f[p_, e_] := Floor[(e + 2)/2] - Floor[(e + 2)/3]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 10000], # > 0 &] (* Amiram Eldar, Feb 23 2024 *)

lista(kmax) = {my(e); for(k = 1, kmax, e = factor(k)[, 2]; if(k == 1 || vecmin(e) > 1, print1(vecprod(apply(x -> (x+2)\2 - (x+2)\3, e)), ", ")));} \\ Amiram Eldar, Feb 23 2024

From Amiram Eldar, Feb 23 2024: (Start)
a(n) = A370256(A001694(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3). (End)

A384790 Numbers with a record number of ways in which they can be expressed as b^2 * c^3, with b and c >= 1.

Original entry on oeis.org

1, 64, 4096, 46656, 2985984, 191102976, 2176782336, 12230590464, 46656000000, 2985984000000, 34012224000000, 191102976000000, 2176782336000000, 139314069504000000, 351298031616000000, 4001504141376000000, 22483074023424000000, 256096265048064000000, 16390160963076096000000

Offset: 1

Amiram Eldar, Jun 10 2025

nonn

Indices of records of A370256.
All the terms are powerful numbers since A370256(1) = 1 and A370256(n) = 0 if n is a nonpowerful number.
The corresponding record values are 1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..61.
Amiram Eldar, n, a(n), A370256(a(n)) for n = 1..61.
Index entries for sequences related to powerful numbers.

Subsequence of A001694 and A025487 (i.e., of A181800).

Cf. A002182 (sixth root), A046055, A057523, A370256, A384791 (cubefull analog).

f[p_, e_] := Floor[(e + 2)/2] - Floor[(e + 2)/3]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; With[{lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]}, sm = -1; seq = {}; Do[s1 = s[lps[[i]]]; If[s1 > sm, sm = s1; AppendTo[seq, lps[[i]]]], {i, 1, Length[lps]}]; seq]

A385418 The number of unordered factorizations of n into powers of primes of the form p^(2^k-1) where p is prime and k >= 0.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 28 2025

First differs from A304327 and A368248 at n = 64.
First differs from A061704 and A362852 at n = 128.
The number of unordered factorizations of n into powers of primes in A036537.

			  n | a(n) | factorizations
  --+------+-------------------------------------------------------------------
  2 |    8 | 2 * 2 * 2, 2^3
  3 |   64 | 2 * 2 * 2 * 2 * 2 * 2, 2 * 2 * 2 * 2^3, 2^3 * 2^3
  4 |  128 | 2 * 2 * 2 * 2 * 2 * 2 * 2, 2 * 2 * 2 * 2 * 2^3, 2 * 2^3 * 2^3, 2^7

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

Cf. A000929, A036537.
Cf. A061704, A304327, A362852, A368248.
Similar sequences: A000688, A046951, A050361, A050377, A188581, A188585, A322885, A370256, A384912, A384913, A384914, A384915, A384916.

T[n_, k_] := T[n, k] = If[k <= n, T[n - k, k] + T[n, 2*k + 1], Boole[n == 0]]; f[p_, e_] := T[e, 1];
a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

t(n, k) = if(k <= n, t(n-k, k) + t(n, 2*k+1), n == 0);
a(n) = vecprod(apply(x -> t(x, 1), factor(n)[,2]));

Multiplicative with a(p^e) = A000929(e).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{k>=2} zeta(2^k-1) = 1.21213028603089660618... .

cp's OEIS Frontend

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

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A384913 The number of unordered factorizations of n into exponentially Fibonacci powers of primes (A115975).

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A384914 The number of unordered factorizations of n into numbers of the form p^(k^2) where p is prime and k >= 0 (A323520).

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A384915 The number of unordered factorizations of n into powers of primes of the form p^e where p is prime and 0 <= e <= p (A074583).

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A384916 The number of unordered factorizations of n into powers of primes of the form p^e where p is prime and 0 <= e < p.

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A057523 Number of ways n-th powerful (1) number can be expressed as b^2*c^3.

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A384790 Numbers with a record number of ways in which they can be expressed as b^2 * c^3, with b and c >= 1.

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