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A166586
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Totally multiplicative sequence with a(p) = p - 2 for prime p.
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21
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1, 0, 1, 0, 3, 0, 5, 0, 1, 0, 9, 0, 11, 0, 3, 0, 15, 0, 17, 0, 5, 0, 21, 0, 9, 0, 1, 0, 27, 0, 29, 0, 9, 0, 15, 0, 35, 0, 11, 0, 39, 0, 41, 0, 3, 0, 45, 0, 25, 0, 15, 0, 51, 0, 27, 0, 17, 0, 57, 0, 59, 0, 5, 0, 33, 0, 65, 0, 21, 0, 69, 0, 71, 0, 9, 0
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OFFSET
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1,5
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LINKS
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FORMULA
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Multiplicative with a(p^e) = (p-2)^e. If n = Product p(k)^e(k) then a(n) = Product (p(k) - 2)^e(k). a(2k) = 0 for k >= 1.
Dirichlet g.f.: 1 / Product_{p prime} (1 - p^(1 - s) + 2*p^(-s)). The Dirichlet inverse is multiplicative with b(p) = 2 - p, b(p^e) = 0, for e > 1. - Álvar Ibeas, Nov 24 2017 [corrected by Vaclav Kotesovec, Feb 10 2023]
Sum_{k=1..n} a(k) ~ c * n^2/2, where c = Product_{primes} (1 - 1/(1 + p*(p-1)/2)) = 0.3049173579282080265466051390930446635010608835584906520231313997... - Vaclav Kotesovec, Feb 10 2023
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MAPLE
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f:= proc(n) local t;
mul((t[1]-2)^t[2], t=ifactors(n)[2])
end proc:
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MATHEMATICA
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a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 2)^fi[[All, 2]])); Table[a[n], {n, 1, 50}] (* G. C. Greubel, Jun 06 2016 *)
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PROG
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(PARI) a(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] -= 2); factorback(f); \\ Michel Marcus, Dec 13 2014
(PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1-p*X+2*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 10 2023
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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EXTENSIONS
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Erroneous formula and program removed by G. C. Greubel, Jun 06 2016
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STATUS
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approved
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