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A055460 Number of primes with odd exponents in the prime power factorization of n!. 6
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 (list; graph; refs; listen; history; text; internal format)
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

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, A007913, A008833.

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

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Last modified July 22 03:22 EDT 2017. Contains 289648 sequences.