A055460 Number of primes with odd exponents in the prime power factorization of n!.

0, 1, 2, 2, 3, 1, 2, 3, 3, 1, 2, 3, 4, 4, 4, 4, 5, 4, 5, 4, 6, 6, 7, 5, 5, 5, 6, 5, 6, 5, 6, 7, 9, 7, 7, 7, 8, 8, 8, 8, 9, 10, 11, 10, 9, 7, 8, 7, 7, 8, 10, 9, 10, 8, 10, 12, 14, 12, 13, 11, 12, 12, 11, 11, 13, 12, 13, 12, 12, 13, 14, 13, 14, 14, 15, 14, 14, 11, 12, 13, 13, 13, 14, 16, 16, 14

Offset

1,3

Comments

The products of the corresponding primes form A055204.

Also, the number of primes dividing the squarefree part of n! (=A055204(n)).

Also, the number of prime factors in the factorization of n! into distinct terms of A050376. See the references in A241289. - Vladimir Shevelev, Apr 16 2014

References

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 (in Russian; MR 2000f: 11097, pp. 3912-3913).

Links

Max Alekseyev, Table of n, a(n) for n = 1..100000

S. Litsyn and V. S. Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.

Formula

a(n) = A001221(A055204(n)). - Max Alekseyev, Oct 19 2014

From Wolfdieter Lang, Nov 06 2021: (Start)

a(n) = A162642(A000142(n)).

a(n) = A000720(n) - A348841(n), (End)

Example

For n = 100, the exponents of primes in the factorization of n! are {97,48,24,16,9,7,5,5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1,1,1}, and there are 17 odd values: {97,9,7,5,5,3,3,1,1,1,1,1,1,1,1,1,1}, so a(100) = 17.
The factorization of 6! into distinct terms of A050376 is 5*9*16 with only one prime, so a(6)=1. - Vladimir Shevelev, Apr 16 2014

Mathematica

Table[Count[FactorInteger[n!][[All, -1]], m_ /; OddQ@ m] - Boole[n == 1], {n, 100}] (* Michael De Vlieger, Feb 05 2017 *)

Prog

(PARI) a(n) = omega(core(n!))

Crossrefs

Cf. A000142, A000720, A007913, A008833, A162642, A348841.

Cf. A249016 (indices of records), A249017 (values of records)

Sequence in context: A166269 A181648 A182910 * A067514 A115323 A089282

Adjacent sequences: A055457 A055458 A055459 * A055461 A055462 A055463

Keyword

nonn

Author

Labos Elemer, Jun 26 2000

Extensions

Edited by Max Alekseyev, Oct 19 2014

Status

approved