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A003966 Möbius transform of A003958. 3
1, 0, 1, 0, 3, 0, 5, 0, 2, 0, 9, 0, 11, 0, 3, 0, 15, 0, 17, 0, 5, 0, 21, 0, 12, 0, 4, 0, 27, 0, 29, 0, 9, 0, 15, 0, 35, 0, 11, 0, 39, 0, 41, 0, 6, 0, 45, 0, 30, 0, 15, 0, 51, 0, 27, 0, 17, 0, 57, 0, 59, 0, 10, 0, 33, 0, 65, 0, 21, 0, 69, 0, 71, 0, 12, 0, 45, 0, 77, 0, 8, 0, 81, 0, 45, 0, 27, 0, 87, 0, 55, 0, 29, 0, 51, 0, 95, 0, 18 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
LINKS
FORMULA
Multiplicative with a(p^e) = (p-2)(p-1)^(e-1). - David W. Wilson, Sep 01 2001
Dirichlet inverse b(n) is multiplicative with b(p^e) = 2-p for prime p and e > 0 (A276833). - Werner Schulte, Oct 25 2018
Sum_{k=1..n} a(k) ~ c * n^2, where c = 2*Pi^2/(105*zeta(3)) = 0.1563923... . - Amiram Eldar, Oct 23 2022
From Vaclav Kotesovec, Feb 11 2023: (Start)
Dirichlet g.f.: 1/zeta(s) * Product_{p prime} 1 / (1 - p^(1-s) + p^(-s)).
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 + (p^(1-s)-2) / (1 - p + p^s)), (with a product that converges for s=2). (End)
MAPLE
A003966 := proc(n) option remember; local pf, p ; if n = 1 then 1; else pf := ifactors(n)[2] ; if nops(pf) = 1 then p := op(1, pf) ; (op(1, p)-2)*(op(1, p)-1)^(op(2, p)-1) ; else mul(procname(op(1, p)^op(2, p)), p=pf) ; end if; end if; end proc:
seq(A003966(n), n=1..100) ; # R. J. Mathar, Jan 07 2011
MATHEMATICA
f[p_, e_] := (p - 2) (p - 1)^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 23 2022 *)
PROG
(PARI) a(n) = {my(f=factor(n)); for (i=1, #f~, p = f[i, 1]; f[i, 1] = (p-2)*(p-1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f); } \\ Michel Marcus, Feb 27 2015
(PARI)
A003958(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1]--); factorback(f);
A003966(n) = sumdiv(n, d, moebius(n/d)*A003958(d)); \\ Antti Karttunen, Oct 24 2018
(PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1-p*X+X)*(1-X))[n], ", ")) \\ Vaclav Kotesovec, Feb 11 2023
CROSSREFS
Sequence in context: A166586 A122274 A340525 * A123931 A058026 A004605
KEYWORD
nonn,mult
AUTHOR
EXTENSIONS
More terms from Antti Karttunen, Oct 24 2018
STATUS
approved

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Last modified March 19 03:33 EDT 2024. Contains 370952 sequences. (Running on oeis4.)