A050377 -id:A050377 - cp's OEIS frontend

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

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.

A330690 Number of ways to factor A108951(n) into "Fermi-Dirac primes" (A050376), where A108951 is fully multiplicative with a(prime(k)) = k-th primorial.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 4, 2, 1, 2, 1, 2, 4, 4, 1, 4, 1, 2, 4, 2, 1, 4, 8, 2, 4, 2, 1, 4, 1, 4, 4, 2, 8, 8, 1, 2, 4, 4, 1, 4, 1, 2, 4, 2, 1, 4, 16, 8, 4, 2, 1, 8, 8, 4, 4, 2, 1, 8, 1, 2, 4, 6, 8, 4, 1, 2, 4, 8, 1, 8, 1, 2, 8, 2, 16, 4, 1, 4, 16, 2, 1, 8, 8, 2, 4, 4, 1, 8, 16, 2, 4, 2, 8, 6, 1, 16, 4, 16, 1, 4, 1, 4, 8

Offset: 1

Antti Karttunen, Dec 28 2019

nonn

a(64) = 6 is the first term which is not a power of 2.

Antti Karttunen, Table of n, a(n) for n = 1..10201
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A034386, A018819, A050376, A050377, A050378, A108951, A329901.

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A018819(n) = if( n<1, n==0, if( n%2, A018819(n-1), A018819(n/2)+A018819(n-1))); \\ From A018819
A050377(n) = factorback(apply(e -> A018819(e), factor(n)[, 2]));
A330690(n) = A050377(A108951(n));

a(n) = A050377(A108951(n)).

a(n) = A050378(A329901(n)).

A050379 Number of ordered factorizations of n into members of A050376.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 5, 1, 2, 2, 6, 1, 5, 1, 5, 2, 2, 1, 10, 2, 2, 3, 5, 1, 6, 1, 10, 2, 2, 2, 14, 1, 2, 2, 10, 1, 6, 1, 5, 5, 2, 1, 22, 2, 5, 2, 5, 1, 10, 2, 10, 2, 2, 1, 18, 1, 2, 5, 18, 2, 6, 1, 5, 2, 6, 1, 32, 1, 2, 5, 5, 2, 6, 1, 22, 6, 2, 1, 18, 2, 2, 2, 10, 1, 18, 2, 5, 2, 2, 2

Offset: 1

Christian G. Bower, Nov 15 1999

nonn

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3 * 3 and 375 = 3 * 5^3 both have prime signature (3,1).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A002033, A050376, A050377, A050378, A050380.

Cf. also A000142, A002110, A023359.

read(transforms) :
L := [1] :
for n from 2 to 100  do
    if isA050376(n) then
        L := [op(L),-1] ;
    else
        L := [op(L),0] ;
    end if;
end do :
a050379 := DIRICHLETi(L) ; # R. J. Mathar, May 26 2017

A064547(n) = {my(f = factor(n)[, 2]); sum(k=1, #f, hammingweight(f[k])); } \\ Michel Marcus, Feb 10 2016
isA050376(n) = ((1==omega(n)) && (1==A064547(n))); \\ Checking that omega(n) is 1 is just an optimization here.
A050379(n) = if(1==n,n,sumdiv(n,d,if(dA050376(n/d)*A050379(d),0))); \\ Antti Karttunen, Oct 20 2017

Dirichlet g.f.: 1/(1-B(s)) where B(s) is D.g.f. of characteristic function of A050376.
a(p^k) = A023359(k), for any prime p.
a(A002110(n)) = A000142(n) = n!.
a(n) = A050380(A101296(n)). - R. J. Mathar, May 26 2017

A296371 Number of integer partitions of n using Jacobsthal numbers.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 5, 6, 7, 9, 10, 11, 13, 15, 17, 19, 21, 23, 26, 30, 33, 36, 40, 44, 49, 54, 58, 63, 69, 75, 82, 89, 95, 103, 112, 120, 129, 138, 147, 158, 170, 182, 194, 207, 221, 236, 252, 267, 283, 301, 319, 339, 360, 380, 402, 426, 450, 475, 501, 527

Offset: 0

Gus Wiseman, Dec 11 2017

nonn

			The a(10) = 7 partitions are (1111111111), (31111111), (331111), (3331), (511111), (5311), (55).

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000041, A001045, A003158, A018819, A050377, A197911.

nn=6;
jac[n_]:=(2^n-(-1)^n)/3;
Table[SeriesCoefficient[Product[1/(1-x^jac[i]),{i,2,nn}],{x,0,n}],{n,jac[nn]}]

A382295 Decimal expansion of the asymptotic mean of the number of ways to factor k into "Fermi-Dirac primes" when k runs over the positive integers.

Original entry on oeis.org

1, 7, 8, 7, 6, 3, 6, 8, 0, 0, 1, 6, 9, 4, 4, 5, 6, 6, 6, 9, 8, 8, 6, 3, 2, 9, 3, 9, 4, 8, 9, 4, 5, 9, 8, 8, 1, 4, 6, 5, 9, 0, 0, 4, 6, 1, 3, 7, 0, 0, 2, 2, 6, 4, 1, 1, 6, 7, 3, 2, 9, 5, 4, 5, 6, 6, 6, 3, 7, 5, 1, 3, 9, 5, 4, 3, 4, 0, 2, 5, 1, 5, 5, 1, 5, 5, 0, 8, 8, 3, 3, 3, 5, 8, 7, 1, 3, 7, 5, 6, 1, 5, 6, 0, 4

Offset: 1

Amiram Eldar, Mar 21 2025

			1.78763680016944566698863293948945988146590046137002...

Cf. A005117 (positions of 1's in A050377), A050377, A082293 (positions of 2's), A330687 (positions of records).

$MaxExtraPrecision = 1500; m = 1500; em = 50; f[x_] := Log[1-x] - Sum[Log[1-x^(2^k)], {k, 0, em}]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x] * Range[0, m]]; RealDigits[Exp[NSum[Indexed[c, k] * PrimeZetaP[k]/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 120][[1]]

default(realprecision, 120); default(parisize, 10000000);
f(x, n) = (1-x) / prod(k = 0, n, (1 - x^(2^k)));
prodeulerrat(f(1/p, 10))

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A050377(k).

Equals Product_{p prime} f(1/p), where f(x) = (1-x) / Product_{k>=0} (1 - x^(2^k)).

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

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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A330690 Number of ways to factor A108951(n) into "Fermi-Dirac primes" (A050376), where A108951 is fully multiplicative with a(prime(k)) = k-th primorial.

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A050379 Number of ordered factorizations of n into members of A050376.

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A296371 Number of integer partitions of n using Jacobsthal numbers.

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A382295 Decimal expansion of the asymptotic mean of the number of ways to factor k into "Fermi-Dirac primes" when k runs over the positive integers.

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