cp's OEIS Frontend

A325036 Difference between product and sum of prime indices of n.

%I A325036 #11 May 26 2023 13:15:18
%S A325036 1,0,0,-1,0,-1,0,-2,0,-1,0,-2,0,-1,1,-3,0,-1,0,-2,2,-1,0,-3,3,-1,2,-2,
%T A325036 0,0,0,-4,3,-1,5,-2,0,-1,4,-3,0,1,0,-2,5,-1,0,-4,8,2,5,-2,0,1,7,-3,6,
%U A325036 -1,0,-1,0,-1,8,-5,9,2,0,-2,7,4,0,-3,0,-1,10,-2,11,3,0,-4,8,-1,0,0,11,-1,8,-3,0,4,14,-2,9
%N A325036 Difference between product and sum of prime indices of n.
%C A325036 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.
%H A325036 Antti Karttunen, <a href="/A325036/b325036.txt">Table of n, a(n) for n = 1..20000</a>
%H A325036 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A325036 a(n) = A003963(n) - A056239(n).
%F A325036 For all n >= 1, a(A325040(n)) = a(A122111(A325040(n))). - _Antti Karttunen_, May 08 2022
%e A325036 The prime indices of 45 are {2,2,3}, with product 12 and sum 7, so a(45) = 5.
%t A325036 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A325036 Table[Times@@primeMS[n]-Total[primeMS[n]],{n,100}]
%t A325036 dps[n_]:=Module[{pi=Flatten[Table[PrimePi[#[[1]]],#[[2]]]&/@FactorInteger[n]]},Times@@pi-Total[pi]]; Join[{1},Array[dps,100,2]] (* _Harvey P. Dale_, May 26 2023 *)
%o A325036 (PARI)
%o A325036 A003963(n) = { n=factor(n); n[, 1]=apply(primepi, n[, 1]); factorback(n) }; \\ From A003963
%o A325036 A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); };
%o A325036 A325036(n) = (A003963(n) - A056239(n)); \\ _Antti Karttunen_, May 08 2022
%Y A325036 Positions of zeros are A301987. Positions of ones are A325041. Positions of negative ones are A325042.
%Y A325036 Cf. A000720, A001222, A003963, A056239, A112798, A122111, A178503, A175508, A319000.
%Y A325036 Cf. A325032, A325033, A325034, A325035, A325037, A325038, A325040, A325044.
%K A325036 sign
%O A325036 1,8
%A A325036 _Gus Wiseman_, Mar 25 2019
%E A325036 Data section extended up to a(93) by _Antti Karttunen_, May 08 2022