|
|
A318762
|
|
Number of permutations of a multiset whose multiplicities are the prime indices of n.
|
|
23
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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.
|
|
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])):
|
|
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.
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|