cp's OEIS Frontend

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.

%I A385418 #8 Jun 29 2025 10:09:34
%S A385418 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,
%T A385418 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,
%U A385418 1,1,1,2,1,1,1,1,1,1,1,2,2,1,1,1,1,1,1
%N 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.
%C A385418 First differs from A304327 and A368248 at n = 64.
%C A385418 First differs from A061704 and A362852 at n = 128.
%C A385418 The number of unordered factorizations of n into powers of primes in A036537.
%H A385418 Amiram Eldar, <a href="/A385418/b385418.txt">Table of n, a(n) for n = 1..10000</a>
%F A385418 Multiplicative with a(p^e) = A000929(e).
%F A385418 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{k>=2} zeta(2^k-1) = 1.21213028603089660618... .
%e A385418   n | a(n) | factorizations
%e A385418   --+------+-------------------------------------------------------------------
%e A385418   2 |    8 | 2 * 2 * 2, 2^3
%e A385418   3 |   64 | 2 * 2 * 2 * 2 * 2 * 2, 2 * 2 * 2 * 2^3, 2^3 * 2^3
%e A385418   4 |  128 | 2 * 2 * 2 * 2 * 2 * 2 * 2, 2 * 2 * 2 * 2 * 2^3, 2 * 2^3 * 2^3, 2^7
%t A385418 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];
%t A385418 a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A385418 (PARI) t(n, k) = if(k <= n, t(n-k, k) + t(n, 2*k+1), n == 0);
%o A385418 a(n) = vecprod(apply(x -> t(x, 1), factor(n)[,2]));
%Y A385418 Cf. A000929, A036537.
%Y A385418 Cf. A061704, A304327, A362852, A368248.
%Y A385418 Similar sequences: A000688, A046951, A050361, A050377, A188581, A188585, A322885, A370256, A384912, A384913, A384914, A384915, A384916.
%K A385418 nonn,easy,mult
%O A385418 1,8
%A A385418 _Amiram Eldar_, Jun 28 2025