A249017 -id:A249017 - cp's OEIS frontend

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

Labos Elemer, Jun 26 2000

nonn

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

			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

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).

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.

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

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

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

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

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)

Edited by Max Alekseyev, Oct 19 2014

A249016 Indices of records in A055460 (number of primes dividing the squarefree part of n!).

Original entry on oeis.org

1, 2, 3, 5, 13, 17, 21, 23, 33, 42, 43, 56, 57, 75, 84, 99, 101, 105, 109, 119, 133, 139, 157, 162, 163, 182, 183, 207, 208, 219, 220, 255, 257, 263, 267, 303, 305, 307, 315, 340, 341, 343, 383, 385, 387, 397, 411, 420, 421, 423, 469, 483, 485, 489, 505, 519, 523, 547, 552, 553, 581, 602, 603, 609, 618

Offset: 1

Max Alekseyev, Oct 19 2014

nonn

Max Alekseyev, Table of n, a(n) for n = 1..38282

Cf. A249017 (values of records), A055204

np=vector(10^6); b=-1; r=0; for(n=1, 10^6, f=factor(n); for(i=1, matsize(f)[1], if(f[i, 2]%2, r += (-1)^np[f[i, 1]]; np[f[i, 1]]=1-np[f[i, 1]]; )); if(r>b, b=r; print1(n, ", ")))

cp's OEIS Frontend

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

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