A318762 Number of permutations of a multiset whose multiplicities are the prime indices of n.

1, 1, 1, 2, 1, 3, 1, 6, 6, 4, 1, 12, 1, 5, 10, 24, 1, 30, 1, 20, 15, 6, 1, 60, 20, 7, 90, 30, 1, 60, 1, 120, 21, 8, 35, 180, 1, 9, 28, 120, 1, 105, 1, 42, 210, 10, 1, 360, 70, 140, 36, 56, 1, 630, 56, 210, 45, 11, 1, 420, 1, 12, 420, 720, 84, 168, 1, 72, 55

Offset: 1

Author

Gus Wiseman, Sep 03 2018

Keywords

• nonn
• look

Comments

This multiset is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

Examples

			The a(12) = 12 permutations are (1123), (1132), (1213), (1231), (1312), (1321), (2113), (2131), (2311), (3112), (3121), (3211).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000041, A000110, A000258, A000670, A005651, A008277, A008480, A056239, A112624, A112798, A124794, A215366, A300335, A305936.

Programs

Maple

a:= n-> (l-> add(i, i=l)!/mul(i!, i=l))(map(i->
       numtheory[pi](i[1])$i[2], ifactors(n)[2])):
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 03 2018

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[primeMS[n]]!/Times@@Factorial/@primeMS[n],{n,100}]

PARI

sig(n)={my(f=factor(n)); concat(vector(#f~, i, vector(f[i, 2], j, primepi(f[i, 1]))))}
a(n)={if(n==1, 1, my(s=sig(n)); vecsum(s)!/prod(i=1, #s, s[i]!))}  \\ Andrew Howroyd, Dec 17 2018

Formula

If n = Product prime(x_i)^y_i is the prime factorization of n, then a(n) = (Sum x_i * y_i)! / Product (x_i!)^y_i.

a(n) = A008480(A181821(n)).

a(n) = A112624(n) * A124794(n). - Max Alekseyev, Oct 15 2023

Sum_{m in row n of A215366} a(m) = A005651(n).

Sum_{m in row n of A215366} a(m) * A008480(m) = A000670(n).

Sum_{m in row n of A215366} a(m) * A008480(m) / A001222(m)! = A000110(n).