A120007 Mobius transform of sum of prime factors of n with multiplicity (A001414).

0, 2, 3, 2, 5, 0, 7, 2, 3, 0, 11, 0, 13, 0, 0, 2, 17, 0, 19, 0, 0, 0, 23, 0, 5, 0, 3, 0, 29, 0, 31, 2, 0, 0, 0, 0, 37, 0, 0, 0, 41, 0, 43, 0, 0, 0, 47, 0, 7, 0, 0, 0, 53, 0, 0, 0, 0, 0, 59, 0, 61, 0, 0, 2, 0, 0, 67, 0, 0, 0, 71, 0, 73, 0, 0, 0, 0, 0, 79, 0, 3, 0, 83, 0, 0, 0, 0, 0, 89, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Franklin T. Adams-Watters, Jun 02 2006

Keywords

• nonn

Comments

Same as A014963, except this function is zero when n is not a prime power, whereas A014963 is one.

Moreover, this sequence, A014963, A297108 and A297109 partition the natural numbers to identical equivalence classes: For all i, j >= 1, a(i) = a(j) <=> A014963(i) = A014963(j) <=> A297108(i) = A297108(j) <=> A297109(i) = A297109(j). - Antti Karttunen, Feb 01 2021

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Prime Factor.
• Eric Weisstein's World of Mathematics, Prime Zeta Function.

Crossrefs

Cf. A000040, A001414, A007947, A014963, A010051, A010055, A061397, A070939, A140508 (Möbius transform of this sequence), A297108, A297109.

Programs

Haskell

a120007 1 = 0
a120007 n | until ((> 0) . (`mod` spf)) (`div` spf) n == 1 = spf
          | otherwise = 0
          where spf = a020639 n
-- Reinhard Zumkeller, Sep 19 2011

Mathematica

Table[If[Length@ # == 1, #[[1, 1]], 0] &@ FactorInteger@ n, {n, 96}] /. 1 -> 0 (* Michael De Vlieger, Jun 19 2016 *)
Table[If[PrimePowerQ[n],FactorInteger[n][[1,1]],0],{n,100}] (* Harvey P. Dale, Jan 25 2020 *)

PARI

A120007(n) = { my(v); if(isprimepower(n, &v), v, 0); }; \\ Antti Karttunen, Jan 31 2021

Formula

If n is a prime power p^k, k>0, a(n) = p; otherwise a(n) = 0.

Dirichlet g.f. sum_{p prime} p/(p^s-1) = sum_{k>0} primezeta(ks-1).

a(n) = A010055(n) * A007947(n). - Reinhard Zumkeller, Mar 26 2010

a(n) = A061397(A007947(n)). - Reinhard Zumkeller, Sep 19 2011, corrected by Antti Karttunen, Jan 31 2021

a(n) = Sum_{k=2..n} k*A010051(k)*(floor(k^n/n)-floor((k^n -1)/n)). - Anthony Browne, Jun 17 2016

If A297109(n) = 0, then a(n) = 0, otherwise a(n) = A000040(A297109(n)). - Antti Karttunen, Feb 01 2021