A385378 -id:A385378 - cp's OEIS frontend

A024923 Partial products of the sequence of prime powers (A000961).

1, 2, 6, 24, 120, 840, 6720, 60480, 665280, 8648640, 138378240, 2352430080, 44696171520, 1028011944960, 25700298624000, 693908062848000, 20123333822592000, 623823348500352000, 19962347152011264000, 738606844624416768000, 30282880629601087488000, 1302163867072846761984000

Offset: 1

Den Roussel (DenRoussel(AT)webtv.net)

nonn

Amiram Eldar, Table of n, a(n) for n = 1..361

Cf. A000961, A024918.
Subsequence of A025487.
The distinct terms in A308819.
Indices of records in A385378.

FoldList[Times, 1, Select[Range[50], PrimePowerQ]] (* Amiram Eldar, Jun 27 2025 *)

ispp1(n) = isprimepower(n) || (n==1); \\ A000961
lista(nn) = {my(s=1); for (n=1, nn, if (ispp1(n), s*= n; print1(s, ", ")););} \\ Michel Marcus, Mar 26 2020

Offset 1 and more terms from Michel Marcus, Mar 26 2020

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

Original entry on oeis.org

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

Amiram Eldar, Jun 27 2025

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).

			         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

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

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

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

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

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.

cp's OEIS Frontend

A024923 Partial products of the sequence of prime powers (A000961).

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