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A120007 Mobius transform of sum of prime factors of n with multiplicity (A001414). 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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
Eric Weisstein's World of Mathematics, Prime Factor.
Eric Weisstein's World of Mathematics, Prime Zeta Function.
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
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 *)
PROG
(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
(PARI) A120007(n) = { my(v); if(isprimepower(n, &v), v, 0); }; \\ Antti Karttunen, Jan 31 2021
CROSSREFS
Cf. A000040, A001414, A007947, A014963, A010051, A010055, A061397, A070939, A140508 (Möbius transform of this sequence), A297108, A297109.
Sequence in context: A295703 A059051 A130069 * A092509 A214053 A214056
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified March 19 06:53 EDT 2024. Contains 370953 sequences. (Running on oeis4.)