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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.
2
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
1,3
COMMENTS
SEP stands for Signed by Exponents of Prime factors.
By "Max(r_j)" I mean the following: If d|m, d=p^e*q^f, m=p^x*q^y*r^z then Max(e)=x, Max(f)=y.
LINKS
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)
EXAMPLE
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.
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
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 *)
PROG
(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
CROSSREFS
Sequence in context: A051559 A073239 A134882 * A257345 A110637 A092943
KEYWORD
nonn,easy,mult
AUTHOR
Yasutoshi Kohmoto, Jan 12 2007, Jan 29 2007
EXTENSIONS
More terms from R. J. Mathar, Jun 07 2007
Formula clarified by Antti Karttunen, Feb 21 2022
STATUS
approved