A240502 Product of primes appearing in the factorization of n! with even exponents.

1, 1, 1, 1, 1, 1, 6, 6, 3, 3, 30, 30, 10, 10, 35, 21, 21, 21, 42, 42, 210, 10, 55, 55, 330, 330, 2145, 715, 5005, 5005, 6006, 6006, 3003, 91, 3094, 2210, 2210, 2210, 20995, 4845, 1938, 1938, 2261, 2261, 24871, 124355, 5720330, 5720330, 17160990, 17160990, 8580495

Offset: 0

Author

Vladimir Shevelev, Apr 06 2014

Keywords

• nonn

Comments

All terms are squarefree (A005117). - Michel Marcus, Feb 15 2016

Examples

			In the prime power factorization 2^7*3^4*5*7 of 9! only the exponent of 3 is even. Thus a(9)=3.
		

Links

• David A. Corneth, Table of n, a(n) for n = 0..7585

Crossrefs

Cf. A000142, A005117, A055204.

Programs

Mathematica

Table[Times@@Select[FactorInteger[n!],EvenQ[#[[2]]]&][[;;,1]],{n,0,50}] (* Harvey P. Dale, Feb 24 2023 *)

PARI

a(n) = {my(f = factor(n!)); for (k=1, #f~, f[k, 2] = 1 - (f[k, 2] % 2);); factorback(f);} \\ Michel Marcus, Feb 15 2016

PARI

a(n) = {my(res=1); forprime(p=2, n\2, e=val(n,p); if(e%2==0,res*=p)); res}
val(n, p) = my(r=0); while(n, r+=n\=p); r \\ David A. Corneth, Feb 24 2023

Formula

a(n) = rad(n!)/core(n!) = A336643(n!). - Benoit Cloitre, Mar 12 2022

Extensions

More terms from Michel Marcus, Feb 15 2016