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

1,4

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}.

Links

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

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).

Example

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

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}]

Prog

(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

Crossrefs

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

Sequence in context: A340793 A053222 A351159 * A262598 A370899 A191854

Adjacent sequences: A318759 A318760 A318761 * A318763 A318764 A318765

Keyword

nonn,look

Author

Gus Wiseman, Sep 03 2018

Status

approved