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A353507
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Product of multiplicities of the prime exponents (signature) of n; a(1) = 0.
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14
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0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 2, 2, 2, 2
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OFFSET
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1,6
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COMMENTS
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Warning: If the prime multiplicities of n are a multiset y, this sequence gives the product of multiplicities in y, not the product of y.
We set a(1) = 0 so that the positions of first appearances are the primorials A002110.
Also the product of the prime metasignature of n (row n of A238747).
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LINKS
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FORMULA
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EXAMPLE
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The prime signature of 13860 is (2,2,1,1,1), with multiplicities (2,3), so a(13860) = 6.
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MAPLE
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f:= proc(n) local M, s;
M:= ifactors(n)[2][.., 2];
mul(numboccur(s, M), s=convert(M, set));
end proc:
f(1):= 0:
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MATHEMATICA
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Table[If[n==1, 0, Times@@Length/@Split[Sort[Last/@FactorInteger[n]]]], {n, 100}]
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PROG
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(Python)
from math import prod
from itertools import groupby
from sympy import factorint
def A353507(n): return 0 if n == 1 else prod(len(list(g)) for k, g in groupby(factorint(n).values())) # Chai Wah Wu, May 20 2022
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CROSSREFS
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Positions of first appearances are A002110.
The prime indices themselves have product A003963, counted by A339095.
A071625 counts distinct prime exponents (third omega).
A130091 lists numbers with distinct prime exponents, counted by A098859.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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