A125140 SEPSigma(n) = (-1)^(Sum_i r_i)*Sum_{d|n} (-1)^(Sum_j Max(r_j))*d = Product_i (Sum_{s_i=1..r_i} p_i^s_i)+(-1)^r_i where n = Product_i p_i^r_i, d = Product_j p_j^r_j, p_j^max(r_j) is the largest power of p_j dividing n.
1, 1, 2, 7, 4, 2, 6, 13, 13, 4, 10, 14, 12, 6, 8, 31, 16, 13, 18, 28, 12, 10, 22, 26, 31, 12, 38, 42, 28, 8, 30, 61, 20, 16, 24, 91, 36, 18, 24, 52, 40, 12, 42, 70, 52, 22, 46, 62, 57, 31, 32, 84, 52, 38, 40, 78, 36, 28, 58, 56, 60, 30, 78, 127, 48, 20, 66, 112, 44, 24, 70, 169, 72
Offset: 1
Examples
If n = 240, d = 12 then 2^max(r_j) = 2^max(2) = 2^4, 3^max(r_j) = 3^max(1) = 3^1, SEPSigma(240) = (1+2+4+8+16)*(-1+3)*(-1+5) = 248.
Links
- Antti Karttunen, Table of n, a(n) for n = 1..20000
Programs
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Maple
A125140 := proc(n) local ifs,i,a,r,p ; ifs := ifactors(n)[2] ; a := 1 ; for i from 1 to nops(ifs) do r := op(2,op(i,ifs)) ; p := op(1,op(i,ifs)) ; a := a*(p*(1-p^r)/(1-p)+(-1)^r) ; od ; RETURN(a) ; end: for n from 1 to 80 do printf("%d, ",A125140(n)) ; od ; # R. J. Mathar, Jun 07 2007
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Mathematica
f[p_, e_] := (p^(e+1) - p)/(p - 1) + (-1)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2023 *)
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PARI
A125140(n) = { my(f=factor(n), p, e); prod(k=1,#f~,p = f[k,1]; e = f[k,2]; ((-1)^e) + (((p^(e+1))-p) / (p-1))); }; \\ Antti Karttunen, Feb 21 2022
Formula
a(n) = Product_i (-1)^r_i + ((p_i^(r_i+1)-p_i)/(p_i-1)), where p_i and r_i range over the primes and their exponents in the prime factorization of n.
a(n) = Product_{p^e || n} (-1)^e + ((p^(1+e)-p)/(p-1)), where p and e range over the primes and their exponents in the prime factorization of n.
From Amiram Eldar, Sep 18 2023: (Start)
Dirichlet g.f.: zeta(s-1) * zeta(2*s) * Product_{p prime} (1 - 1/p^s + 2/p^(2*s-1)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - (p^2 - 2*p - 1)/(p^4 - 1)) = 0.48777088716109463306... . (End)
Extensions
More terms from R. J. Mathar, Jun 07 2007
Formula clarified by Antti Karttunen, Feb 21 2022
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