A376886 The number of distinct factors of n of the form p^(k!), where p is a prime and k >= 1, when the factorization is uniquely done using the factorial-base representation of the exponents in the prime factorization of n.

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

Offset: 1

Author

Amiram Eldar, Oct 08 2024

Keywords

• nonn
• easy
• base

Comments

See A376885 for details about this factorization.

First differs from A371090 at n = 2^18 = 262144.

Differs from A064547 at n = 64, 128, 192, 256, 320, 384, 448, 512, ... .

Differs from A058061 at n = 128, 384, 512, 640, 896, ... .

Examples

			For n = 8 = 2^3, the representation of 3 in factorial base is 11, i.e., 3 = 1! + 2!, so 8 = (2^(1!))^1 * (2^(2!))^1 and a(8) = 1 + 1 = 2.
For n = 16 = 2^4, the representation of 4 in factorial base is 20, i.e., 4 = 2 * 2!, so 16 = (2^(2!))^2 and a(16) = 1.
		

Links

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

Crossrefs

Cf. A007623, A058061, A060130, A077761, A371090.

Similar sequences: A064547, A318464, A376885.

Programs

Mathematica

fdignum[n_] := Module[{k = n, m = 2, r, s = 0}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, If[r > 0, s++]; m++]; s]; f[p_, e_] := fdignum[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

fdignum(n) = {my(k = n, m = 2, r, s = 0); while([k, r] = divrem(k, m); k != 0 || r != 0, if(r > 0, s ++); m++); s;}
a(n) = {my(e = factor(n)[, 2]); sum(i = 1, #e, fdignum(e[i]));}

Formula

Additive with a(p^e) = A060130(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} f(1/p) = 0.12589120926760155013..., where f(x) = -x + (1-x) * Sum_{k>=1} A060130(k) * x^k.