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 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 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)). 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, A112798, A215366, A300335, A305936. Sequence in context: A205858 A340793 A053222 * A262598 A191854 A129646 Adjacent sequences:  A318759 A318760 A318761 * A318763 A318764 A318765 KEYWORD nonn,look AUTHOR Gus Wiseman, Sep 03 2018 STATUS approved

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Last modified July 28 14:19 EDT 2021. Contains 346335 sequences. (Running on oeis4.)