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A195017
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If n = Product_{k >= 1} (p_k)^(c_k) where p_k is k-th prime and c_k >= 0 then a(n) = Sum_{k >= 1} c_k*((-1)^(k-1)).
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55
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0, 1, -1, 2, 1, 0, -1, 3, -2, 2, 1, 1, -1, 0, 0, 4, 1, -1, -1, 3, -2, 2, 1, 2, 2, 0, -3, 1, -1, 1, 1, 5, 0, 2, 0, 0, -1, 0, -2, 4, 1, -1, -1, 3, -1, 2, 1, 3, -2, 3, 0, 1, -1, -2, 2, 2, -2, 0, 1, 2, -1, 2, -3, 6, 0, 1, 1, 3, 0, 1, -1, 1, 1, 0, 1, 1, 0, -1, -1, 5, -4, 2, 1, 0, 2, 0, -2, 4, -1, 0, -2, 3, 0, 2, 0, 4, 1, -1, -1, 4, -1, 1, 1, 2, -1
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OFFSET
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1,4
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COMMENTS
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Let p(n,x) be the completely additive polynomial-valued function such that p(1,x) = 0 and p(prime(n),x) = x^(n-1), like is defined in A206284 (although here we are not limited to just irreducible polynomials). Then a(n) is the value of the polynomial encoded in such a manner by n, when it is evaluated at x=-1. - The original definition rewritten and clarified by Antti Karttunen, Oct 03 2018
Positions of 0 give the values of n for which the polynomial p(n,x) is divisible by x+1. For related sequences, see the Mathematica section.
Also the number of odd prime indices of n minus the number of even prime indices of n (both counted with multiplicity), where a prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. - Gus Wiseman, Oct 24 2023
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LINKS
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FORMULA
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Totally additive with a(p^e) = e * (-1)^(1+PrimePi(p)), where PrimePi(n) = A000720(n). - Antti Karttunen, Oct 03 2018
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} = (-1)^(primepi(p)+1)/(p-1) = Sum_{k>=1} (-1)^(k+1)/A006093(k) = A078437 + Sum_{k>=1} (-1)^(k+1)/A036689(k) = 0.6339266524059... . - Amiram Eldar, Sep 29 2023
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EXAMPLE
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The sequence can be read from a list of the polynomials:
p(n,x) with x = -1, gives a(n)
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p(1,x) = 0 0
p(2,x) = 1x^0 1
p(3,x) = x -1
p(4,x) = 2x^0 2
p(5,x) = x^2 1
p(6,x) = 1+x 0
p(7,x) = x^3 -1
p(8,x) = 3x^0 3
p(9,x) = 2x -2
p(10,x) = x^2 + 1 2.
(The list runs through all the polynomials whose coefficients are nonnegative integers.)
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MATHEMATICA
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b[n_] := Table[x^k, {k, 0, n}];
f[n_] := f[n] = FactorInteger[n]; z = 200;
t[n_, m_, k_] := If[PrimeQ[f[n][[m, 1]]] && f[n][[m, 1]]
== Prime[k], f[n][[m, 2]], 0];
u = Table[Apply[Plus,
Table[Table[t[n, m, k], {k, 1, PrimePi[n]}], {m, 1,
Length[f[n]]}]], {n, 1, z}];
p[n_, x_] := u[[n]].b[-1 + Length[u[[n]]]]
Table[p[n, x] /. x -> 0, {n, 1, z/2}] (* A007814 *)
Table[p[2 n, x] /. x -> 0, {n, 1, z/2}] (* A001511 *)
Table[p[n, x] /. x -> 1, {n, 1, z}] (* A001222 *)
Table[p[n, x] /. x -> 2, {n, 1, z}] (* A048675 *)
Table[p[n, x] /. x -> 3, {n, 1, z}] (* A090880 *)
Table[p[n, x] /. x -> -1, {n, 1, z}] (* A195017 *)
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PROG
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(PARI) A195017(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * (-1)^(1+primepi(f[i, 1])))); } \\ Antti Karttunen, Oct 03 2018
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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EXTENSIONS
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More terms, name changed and example-section edited by Antti Karttunen, Oct 03 2018
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STATUS
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approved
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