cp's OEIS Frontend

A380344 Product of prime indices minus sum of prime factors of n.

%I A380344 #9 Jan 26 2025 09:12:56
%S A380344 1,-1,-1,-3,-2,-3,-3,-5,-2,-4,-6,-5,-7,-5,-2,-7,-10,-4,-11,-6,-2,-8,
%T A380344 -14,-7,-1,-9,-1,-7,-19,-4,-20,-9,-4,-12,0,-6,-25,-13,-4,-8,-28,-4,
%U A380344 -29,-10,1,-16,-32,-9,2,-3,-6,-11,-37,-3,-1,-9,-6,-21,-42,-6,-43
%N A380344 Product of prime indices minus sum of prime factors of n.
%C A380344 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, with product A003963.
%F A380344 a(n) = A003963(n) - A001414(n).
%e A380344 72 has prime factors {2,2,2,3,3} and prime indices {1,1,1,2,2}, so a(72) = 4 - 12 = -8.
%t A380344 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A380344 Table[Times@@prix[n]-Plus@@Prime/@prix[n],{n,100}]
%Y A380344 Positions of 0 are A331384.
%Y A380344 For plus instead of minus we have A380409.
%Y A380344 Positions of positives are A380410.
%Y A380344 Triangles:
%Y A380344 - A027746 = prime factors
%Y A380344 - A112798 = prime indices
%Y A380344 Statistics:
%Y A380344 - A000027 = product of prime factors = row products of A027746
%Y A380344 - A001414 = sum of prime factors = row sums of A027746
%Y A380344 - A003963 = product of prime indices = row products of A112798
%Y A380344 - A056239 = sum of prime indices = row sums of A112798
%Y A380344 Combinations:
%Y A380344 - A075254 = product of factors + sum of factors = A000027 + A001414
%Y A380344 - A075255 = product of factors - sum of factors = A000027 - A001414
%Y A380344 - A178503 = product of factors - sum of indices = A000027 - A056239
%Y A380344 - A325036 = product of indices - sum of indices = A003963 - A056239
%Y A380344 - A379681 = product of indices + sum of indices = A003963 + A056239
%Y A380344 - A380344 = product of indices - sum of factors = A003963 - A001414
%Y A380344 - A380345 = product of factors + sum of indices = A000027 + A056239
%Y A380344 - A380409 = product of indices + sum of factors = A003963 + A001414
%Y A380344 A000040 lists the primes, differences A001223.
%Y A380344 A001222 counts prime factors with multiplicity.
%Y A380344 A055396 gives least prime index, greatest A061395.
%Y A380344 Cf. A000720, A175508, A319000, A325032, A325033, A325034, A325035, A325040, A379682, A380220.
%K A380344 sign
%O A380344 1,4
%A A380344 _Gus Wiseman_, Jan 24 2025