A293902 If n = p_1^e_1 * ... * p_k^e_k, p_1, ..., p_k primes, then a(n) = Product c! where c ranges over products of all combinations of exponents e_1, ..., e_k as {e_1, e_1*e_2, e_1*e_3, e_2*e_3, e_1*e_2*e_3, ..., e_1*e_2*...*e_k}.

1, 1, 1, 2, 1, 1, 1, 6, 2, 1, 1, 4, 1, 1, 1, 24, 1, 4, 1, 4, 1, 1, 1, 36, 2, 1, 6, 4, 1, 1, 1, 120, 1, 1, 1, 96, 1, 1, 1, 36, 1, 1, 1, 4, 4, 1, 1, 576, 2, 4, 1, 4, 1, 36, 1, 36, 1, 1, 1, 16, 1, 1, 4, 720, 1, 1, 1, 4, 1, 1, 1, 8640, 1, 1, 4, 4, 1, 1, 1, 576, 24, 1, 1, 16, 1, 1, 1, 36, 1, 16, 1, 4, 1, 1, 1, 14400, 1, 4, 4, 96, 1, 1, 1, 36, 1

Offset: 1

Author

Antti Karttunen, Oct 22 2017

Keywords

• nonn

Comments

a(1) = 1 (an empty product).

Examples

			For n = 36 = 2^2 * 3^2 the combinations of the exponents are [], [2] (as exponent of 2), [2] (as exponent of 3) and [2, 2]. Taking products of these multisets we get 1 (as an empty product), 2, 2 and 4. Thus a(36) = 1! * 2! * 2! * 4! = 1*2*2*24 = 96.
For n = 72 = 2^3 * 3^2 the combinations of the exponents are [], [2], [3] and [2, 3]. Taking products of these multisets we get 1, 2, 3 and 6. Thus a(72) = 1! * 2! * 3! * 6! = 1*2*6*720 = 8640.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A000142, A163820, A293900.

Programs

Mathematica

Array[Apply[Times, Map[Times @@ # &, Subsets@ FactorInteger[#][[All, -1]]]!] &, 105] (* Michael De Vlieger, Oct 23 2017 *)

PARI

A293902(n) = { my(exp_combos=powerset(factor(n)[, 2]), m=1); for(i=1,#exp_combos,m *= vecproduct(exp_combos[i])!); m; };
vecproduct(v) = { my(m=1); for(i=1,#v,m *= v[i]); m; };
powerset(v) = { my(siz=2^length(v),pv=vector(siz)); for(i=0,siz-1,pv[i+1] = choosebybits(v,i)); pv; };
choosebybits(v,m) = { my(s=vector(hammingweight(m)),i=j=1); while(m>0,if(m%2,s[j] = v[i];j++); i++; m >>= 1); s; };

Scheme

(define (A293902 n) (if (= 1 n) n (/ (A163820 n) (A293900 n))))

Formula

For n = p^k * q * ... * r (with only one of the prime factors occurring multiple times), a(n) = A000142(k)^(2^(A001221(n)-1)).

a(p^n) = A000142(n), for any prime p.

For n > 1, a(n) = A163820(n) / A293900(n).