cp's OEIS Frontend

A380409 Product of prime indices plus sum of prime factors.

%I A380409 #5 Jan 26 2025 09:12:52
%S A380409 1,3,5,5,8,7,11,7,10,10,16,9,19,13,14,9,24,12,27,12,18,18,32,11,19,21,
%T A380409 17,15,39,16,42,11,24,26,24,14,49,29,28,14,54,20,57,20,23,34,62,13,30,
%U A380409 21,34,23,69,19,31,17,38,41,76,18,79,44,29,13,36,26,86
%N A380409 Product of prime indices plus sum of prime factors.
%C A380409 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 A380409 a(n) = A003963(n) + A001414(n).
%e A380409 72 has prime factors {2,2,2,3,3} and prime indices {1,1,1,2,2}, so a(72) = 12 + 4 = 16.
%t A380409 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A380409 Table[Total[Prime/@prix[n]]+Times@@prix[n],{n,100}]
%Y A380409 For factors instead of indices we have A075254.
%Y A380409 For indices instead of factors we have A379681.
%Y A380409 For minus instead of plus we have A380344, zeros A331384.
%Y A380409 Triangles:
%Y A380409 - A027746 = prime factors
%Y A380409 - A112798 = prime indices
%Y A380409 Statistics:
%Y A380409 - A000027 = product of prime factors = row products of A027746
%Y A380409 - A001414 = sum of prime factors = row sums of A027746
%Y A380409 - A003963 = product of prime indices = row products of A112798
%Y A380409 - A056239 = sum of prime indices = row sums of A112798
%Y A380409 Combinations:
%Y A380409 - A075254 = product of factors + sum of factors = A000027 + A001414
%Y A380409 - A075255 = product of factors - sum of factors = A000027 - A001414
%Y A380409 - A178503 = product of factors - sum of indices = A000027 - A056239
%Y A380409 - A325036 = product of indices - sum of indices = A003963 - A056239
%Y A380409 - A379681 = product of indices + sum of indices = A003963 + A056239
%Y A380409 - A380344 = product of indices - sum of factors = A003963 - A001414
%Y A380409 - A380345 = product of factors + sum of indices = A000027 + A056239
%Y A380409 - A380409 = product of indices + sum of factors = A003963 + A001414
%Y A380409 A000040 lists the primes, differences A001223.
%Y A380409 A001222 counts prime factors with multiplicity.
%Y A380409 A055396 gives least prime index, greatest A061395.
%Y A380409 Cf. A000720, A175508, A319000, A325032, A325033, A325034, A325035, A325040, A379682, A380220, A380410.
%K A380409 nonn
%O A380409 1,2
%A A380409 _Gus Wiseman_, Jan 25 2025