

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 FermiDirac arithmetic, Izvestia Vuzov of the NorthCaucasus region, Nature sciences 4 (1996), 2843 (in Russian; MR 2000f: 11097, pp. 39123913).


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, 136.


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



