cp's OEIS Frontend

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

%I A120007 #35 Feb 16 2025 08:33:01
%S A120007 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,
%T A120007 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,
%U A120007 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
%N A120007 Mobius transform of sum of prime factors of n with multiplicity (A001414).
%C A120007 Same as A014963, except this function is zero when n is not a prime power, whereas A014963 is one.
%C A120007 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
%H A120007 Reinhard Zumkeller, <a href="/A120007/b120007.txt">Table of n, a(n) for n = 1..10000</a>
%H A120007 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeFactor.html">Prime Factor.</a>
%H A120007 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeZetaFunction.html">Prime Zeta Function.</a>
%F A120007 If n is a prime power p^k, k>0, a(n) = p; otherwise a(n) = 0.
%F A120007 Dirichlet g.f. sum_{p prime} p/(p^s-1) = sum_{k>0} primezeta(ks-1).
%F A120007 a(n) = A010055(n) * A007947(n). - _Reinhard Zumkeller_, Mar 26 2010
%F A120007 a(n) = A061397(A007947(n)). - _Reinhard Zumkeller_, Sep 19 2011, corrected by _Antti Karttunen_, Jan 31 2021
%F A120007 a(n) = Sum_{k=2..n} k*A010051(k)*(floor(k^n/n)-floor((k^n -1)/n)). - _Anthony Browne_, Jun 17 2016
%F A120007 If A297109(n) = 0, then a(n) = 0, otherwise a(n) = A000040(A297109(n)). - _Antti Karttunen_, Feb 01 2021
%t A120007 Table[If[Length@ # == 1, #[[1, 1]], 0] &@ FactorInteger@ n, {n, 96}] /. 1 -> 0 (* _Michael De Vlieger_, Jun 19 2016 *)
%t A120007 Table[If[PrimePowerQ[n],FactorInteger[n][[1,1]],0],{n,100}] (* _Harvey P. Dale_, Jan 25 2020 *)
%o A120007 (Haskell)
%o A120007 a120007 1 = 0
%o A120007 a120007 n | until ((> 0) . (`mod` spf)) (`div` spf) n == 1 = spf
%o A120007           | otherwise = 0
%o A120007           where spf = a020639 n
%o A120007 -- _Reinhard Zumkeller_, Sep 19 2011
%o A120007 (PARI) A120007(n) = { my(v); if(isprimepower(n, &v), v, 0); }; \\ _Antti Karttunen_, Jan 31 2021
%Y A120007 Cf. A000040, A001414, A007947, A014963, A010051, A010055, A061397, A070939, A140508 (Möbius transform of this sequence), A297108, A297109.
%K A120007 nonn
%O A120007 1,2
%A A120007 _Franklin T. Adams-Watters_, Jun 02 2006