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A380986 Product of prime indices of n (with multiplicity) minus product of distinct prime indices of n.

%I A380986 #7 Feb 18 2025 15:34:33
%S A380986 0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,6,0,6,0,0,0,0,0,0,0,
%T A380986 0,2,0,0,0,0,0,0,0,0,6,0,0,0,12,6,0,0,0,6,0,0,0,0,0,0,0,0,8,0,0,0,0,0,
%U A380986 0,0,0,2,0,0,12,0,0,0,0,0,14,0,0,0,0,0
%N A380986 Product of prime indices of n (with multiplicity) minus product of distinct prime indices of n.
%C A380986 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%F A380986 a(n) = A003963(n) - A156061(n).
%e A380986 The prime indices of 300 are {1,1,2,3,3}, so a(300) = 18 - 6 = 12.
%t A380986 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A380986 Table[Times@@prix[n]-Times@@Union[prix[n]],{n,100}]
%Y A380986 Positions of nonzeros are A038838.
%Y A380986 For length instead of product we have A046660.
%Y A380986 For factors instead of indices we have A066503, see A007947 (squarefree kernel).
%Y A380986 For sum of factors instead of product of indices we have A280292, see A280286, A381075.
%Y A380986 For quotient instead of difference we have A290106, for factors A003557.
%Y A380986 For sum instead of product we have A380955 (firsts A380956, sorted A380957).
%Y A380986 A000040 lists the primes, differences A001223.
%Y A380986 A003963 gives product of prime indices, distinct A156061.
%Y A380986 A005117 lists the squarefree numbers, complement A013929.
%Y A380986 A055396 gives least prime index, greatest A061395.
%Y A380986 A056239 adds up prime indices, row sums of A112798, length A001222.
%Y A380986 A304038 lists distinct prime indices, sum A066328, length A001221.
%Y A380986 Cf. A000720, A081770, A075255, A116861, A178503, A374248, A379681, A380958.
%K A380986 nonn
%O A380986 1,9
%A A380986 _Gus Wiseman_, Feb 14 2025