A003966 Möbius transform of A003958.
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
Offset: 1
Links
- Antti Karttunen, Table of n, a(n) for n = 1..16384
- Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Programs
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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
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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 *)
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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
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PARI
for(n=1, 100, print1(direuler(p=2, n, 1/(1-p*X+X)*(1-X))[n], ", ")) \\ Vaclav Kotesovec, Feb 11 2023
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)
Extensions
More terms from Antti Karttunen, Oct 24 2018