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A353507 Product of multiplicities of the prime exponents (signature) of n; a(1) = 0.

%I A353507 #15 Oct 20 2024 16:39:11
%S A353507 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,
%T A353507 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,
%U A353507 2,3,1,1,1,2,1,1,2,3,1,1,1,2,1,2,2,2,2
%N A353507 Product of multiplicities of the prime exponents (signature) of n; a(1) = 0.
%C A353507 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.
%C A353507 Differs from A351946 at A351946(1260) = 4, a(1260) = 2.
%C A353507 Differs from A327500 at A327500(450) = 3, a(450) = 2.
%C A353507 We set a(1) = 0 so that the positions of first appearances are the primorials A002110.
%C A353507 Also the product of the prime metasignature of n (row n of A238747).
%H A353507 Robert Israel, <a href="/A353507/b353507.txt">Table of n, a(n) for n = 1..10000</a>
%F A353507 a(n) = A005361(A181819(n)) = A003963(A181819(A181819(n))).
%e A353507 The prime signature of 13860 is (2,2,1,1,1), with multiplicities (2,3), so a(13860) = 6.
%p A353507 f:= proc(n) local M,s;
%p A353507   M:= ifactors(n)[2][..,2];
%p A353507   mul(numboccur(s,M),s=convert(M,set));
%p A353507 end proc:
%p A353507 f(1):= 0:
%p A353507 map(f, [$1..100]); # _Robert Israel_, May 19 2023
%t A353507 Table[If[n==1,0,Times@@Length/@Split[Sort[Last/@FactorInteger[n]]]],{n,100}]
%t A353507 Join[{0},Table[Times@@(Length/@Split[FactorInteger[n][[;;,2]]]),{n,2,100}]] (* _Harvey P. Dale_, Oct 20 2024 *)
%o A353507 (Python)
%o A353507 from math import prod
%o A353507 from itertools import groupby
%o A353507 from sympy import factorint
%o A353507 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
%Y A353507 Positions of first appearances are A002110.
%Y A353507 The prime indices themselves have product A003963, counted by A339095.
%Y A353507 The prime signature itself has product A005361, counted by A266477.
%Y A353507 A001222 counts prime factors with multiplicity, distinct A001221.
%Y A353507 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A353507 A071625 counts distinct prime exponents (third omega).
%Y A353507 A124010 gives prime signature, sorted A118914.
%Y A353507 A130091 lists numbers with distinct prime exponents, counted by A098859.
%Y A353507 A181819 gives prime shadow, with an inverse A181821.
%Y A353507 A238747 gives prime metasignature, sorted A353742.
%Y A353507 A323022 gives fourth omega.
%Y A353507 Cf. A085629, A097318, A182850, A304678, A353394, A353399, A353500, A353503.
%K A353507 nonn
%O A353507 1,6
%A A353507 _Gus Wiseman_, May 19 2022