A385379 The maximum possible number of distinct composite prime powers (A246547) in the factorization of n into prime powers.

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

Offset

1,32

Comments

Differs from A376679 at n = 1, 48, 72, 80, ... .

The factorization includes primes if n is not a powerful number (A001694) that is larger than 1.

a(n) depends only on the prime signature of n (A118914).

Links

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

Formula

Additive with a(p^e) = A052146(e+1).

a(n) = 0 if and only if n is squarefree (A005117).

a(A385380(n)) = n-1.

Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761), C = Sum_{k>=1} P(k*(k+3)/2) = 0.49006911093767425812..., and P is the prime zeta function.

Example

         n | a(n) | factorization
  ---------+------+----------------------------------------
         4 |  1   | 2^2
        32 |  2   | 2^2 * 2^3
       288 |  3   | 2^2 * 2^3 * 3^2
      4608 |  4   | 2^2 * 2^3 * 3^2 * 2^4
    115200 |  5   | 2^2 * 2^3 * 3^2 * 2^4 * 5^2
   3110400 |  6   | 2^2 * 2^3 * 3^2 * 2^4 * 5^2 * 3^3
  99532800 |  7   | 2^2 * 2^3 * 3^2 * 2^4 * 5^2 * 3^3 * 2^5

Mathematica

f[p_, e_] := Floor[(Sqrt[8*e + 9] - 3)/2]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = vecsum(apply(x -> (sqrtint(8*x+9)-1)\2 , factor(n)[, 2]));

Crossrefs

Cf. A001694, A005117, A052146, A077761, A118914, A246547, A246655, A376679, A385378, A385380 (indices of records).

Sequence in context: A130207 A325433 A167688 * A376679 A083914 A083891

Adjacent sequences: A385376 A385377 A385378 * A385380 A385381 A385382

Keyword

nonn,easy

Author

Amiram Eldar, Jun 27 2025

Status

approved