A385379 -id:A385379 - cp's OEIS frontend

A385378 The maximum possible number of distinct factors in the factorization of n into prime powers (A246655).

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

Offset: 1

Amiram Eldar, Jun 27 2025

Differs from A376885 and A384422 at n = 32, 64, 96, 128, 160, 192, ... .
Differs from A086435 at n = 36, 100, 144, 180, 196, 225, ... .
Differs from A375272 at n = 128, 384, 640, 896, 1024, 1152, ...	.
a(n) depends only on the prime signature of n (A118914).
The indices of records in this sequence are the partial products of the sequence of powers of primes (A000961), i.e., the terms in A024923.
The least index n such that a(n) = k, for k = 0, 1, 2, ..., is A024923(k+1).

			      n | a(n) | factorization
  ------+------+--------------------------------
      2 |  1   | 2
      6 |  2   | 2 * 3
     24 |  3   | 2 * 3 * 2^2
    120 |  4   | 2 * 3 * 2^2 * 5
    840 |  5   | 2 * 3 * 2^2 * 5 * 7
   6720 |  6   | 2 * 3 * 2^2 * 5 * 7 * 2^3
  60480 |  7   | 2 * 3 * 2^2 * 5 * 7 * 2^3 * 3^2

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

Cf. A000430, A000961, A001221, A003056, A004709, A024923, A077761, A086435, A118914, A246655, A254578, A375272, A376885, A384422, A385379.

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

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

Additive with a(p^e) = A003056(e).
a(n) >= A001221(n), with equality if and only if n is cubefree (A004709).
a(n) >= 1 for n >= 2, with equality if and only if n is a prime or a square of a prime (A000430).
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761), C = Sum_{k>=2} P(k*(k+1)/2) = 0.19285739770001405035..., and P is the prime zeta function.

A385380 Partial products of the sequence nonprime powers of primes (A025475).

Original entry on oeis.org

1, 4, 32, 288, 4608, 115200, 3110400, 99532800, 4877107200, 312134860800, 25282923724800, 3059233770700800, 382404221337600000, 48947740331212800000, 8272168115974963200000, 2010136852181916057600000, 514595034158570510745600000, 148717964871826877605478400000

Offset: 1

Amiram Eldar, Jun 27 2025

Indices of records in A385379.

a(n) is the least index k such that A385379(k) = n-1.

Amiram Eldar, Table of n, a(n) for n = 1..217
Index entries for sequences related to powerful numbers.

Cf. A024923, A025475, A385379.

Subsequence of A001694 and A025487 (i.e., of A181800).

FoldList[Times, 1, Select[Range[250], !PrimeQ[#] && PrimePowerQ[#] &]] (* Amiram Eldar, Jun 27 2025 *)

list(lim) = {my(p = 1); print1(p, ", "); for(k = 2, lim, if(isprimepower(k) > 1, p *= k; print1(p, ", ")));}

a(n) = Product_{k=1..n} A025475(k).

cp's OEIS Frontend

A385378 The maximum possible number of distinct factors in the factorization of n into prime powers (A246655).

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