A325036 - OEIS

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A325036

Difference between product and sum of prime indices of n.

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, 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, -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

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OFFSET

1,8

COMMENTS

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.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000

Index entries for sequences computed from indices in prime factorization

FORMULA

a(n) = A003963(n) - A056239(n).

For all n >= 1, a(A325040(n)) = a(A122111(A325040(n))). - Antti Karttunen, May 08 2022

EXAMPLE

The prime indices of 45 are {2,2,3}, with product 12 and sum 7, so a(45) = 5.

MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Table[Times@@primeMS[n]-Total[primeMS[n]], {n, 100}]

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 *)

PROG

(PARI)

A003963(n) = { n=factor(n); n[, 1]=apply(primepi, n[, 1]); factorback(n) }; \\ From A003963

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); };

A325036(n) = (A003963(n) - A056239(n)); \\ Antti Karttunen, May 08 2022

CROSSREFS

Positions of zeros are A301987. Positions of ones are A325041. Positions of negative ones are A325042.

Cf. A000720, A001222, A003963, A056239, A112798, A122111, A178503, A175508, A319000.

Cf. A325032, A325033, A325034, A325035, A325037, A325038, A325040, A325044.

Sequence in context: A359269 A091603 A370884 * A194942 A129688 A356678

Adjacent sequences: A325033 A325034 A325035 * A325037 A325038 A325039

KEYWORD

sign

AUTHOR

Gus Wiseman, Mar 25 2019

EXTENSIONS

Data section extended up to a(93) by Antti Karttunen, May 08 2022

STATUS

approved