cp's OEIS Frontend

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.

%I A384916 #7 Jun 12 2025 10:24:52
%S A384916 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,
%T A384916 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,
%U A384916 1,1,1,2,1,1,2,1,1,1,1,1,3,1,1,1,1,1,1
%N 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.
%C A384916 First differs from A298735 at n = 125.
%H A384916 Amiram Eldar, <a href="/A384916/b384916.txt">Table of n, a(n) for n = 1..10000</a>
%F A384916 Multiplicative with a(p^e) = A026820(e, p-1).
%F A384916 a(n) <= A384915(n), with equality if and only if n is in A048103.
%e A384916 a(9) = 2 since 9 has 2 factorizations: 3^1 * 3^1 and 3^2, with exponents 1 and 2 that are < 3.
%t A384916 f[p_, e_] := Length[IntegerPartitions[e, p-1]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A384916 (PARI) T(n, k)=my(s); forpart(v=n, s++, , k); s \\ _Charles R Greathouse IV_ at A026820
%o A384916 a(n) = {my(f = factor(n)); prod(i = 1, #f~, T(f[i,2], f[i,1]-1));}
%Y A384916 Cf. A026820, A048103, A298735.	
%Y A384916 Similar sequences: A000688, A046951, A050361, A050377, A188581, A188585, A322885, A370256, A384912, A384913, A384914, A384915.
%K A384916 nonn,easy,mult
%O A384916 1,9
%A A384916 _Amiram Eldar_, Jun 12 2025