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URL: https://oeis.org/A126690

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A126690
Multiplicative function defined for prime powers by a(p^k) = p + p^2 + p^3 + ... + p^(k-1) - 1 (k >= 1).
2
1, -1, -1, 1, -1, 1, -1, 5, 2, 1, -1, -1, -1, 1, 1, 13, -1, -2, -1, -1, 1, 1, -1, -5, 4, 1, 11, -1, -1, -1, -1, 29, 1, 1, 1, 2, -1, 1, 1, -5, -1, -1, -1, -1, -2, 1, -1, -13, 6, -4, 1, -1, -1, -11, 1, -5, 1, 1, -1, 1, -1, 1, -2, 61, 1, -1, -1, -1, 1, -1, -1, 10, -1, 1, -4, -1, 1, -1, -1, -13, 38, 1, -1, 1, 1, 1, 1, -5, -1, 2, 1, -1, 1, 1, 1, -29, -1, -6, -2, 4
OFFSET
1,8
COMMENTS
If we change the definition to a(p^k) = p + p^2 + p^3 + ... + p^k - 1 (k >= 1) we get (-1)sigma(n), A049060.
LINKS
FORMULA
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^(s-1) - 2/p^s + 3/p^(2*s-1)). - Amiram Eldar, May 26 2025
EXAMPLE
a(5) = -1, a(9) = 3-1 = 2, a(45) = (-1)*2 = -2.
MAPLE
pksum := proc(L) local p, k ; p := op(1, L) ; k := op(2, L) ; (p^k-p)/(p-1)-1 ; end: A126690 := proc(n) local pe, a ; if n = 1 then RETURN(1) ; else a := 1 ; pe := ifactors(n)[2] ; for d in pe do a := a*pksum(d) ; od: RETURN(a) ; fi; end: for n from 1 to 120 do printf("%d, ", A126690(n)) ; od: # R. J. Mathar, Aug 08 2008
MATHEMATICA
a[1] = 1;
a[n_] := a[n] = Product[{p, k} = pk; Total[p^Range[k - 1]] - 1, {pk, FactorInteger[n]}];
Array[a, 100] (* Jean-François Alcover, Mar 31 2020 *)
PROG
(Scheme) ;; With memoization-macro definec.
(definec (A126690 n) (cond ((= 1 n) n) ((= 1 (A067029 n)) (- (A126690 (A028234 n)))) (else (* (+ -1 (add (lambda (k) (expt (A020639 n) k)) 1 (- (A067029 n) 1))) (A126690 (A028234 n))))))
;; Auxiliary function add implements sum_{i=lowlim..uplim} intfun(i)
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))
;; Antti Karttunen, Sep 23 2017
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^f[i, 2] - 2*f[i, 1] + 1)/(f[i, 1]-1)); } \\ Amiram Eldar, May 26 2025
CROSSREFS
Cf. A049060.
Sequence in context: A326327 A113103 A033325 * A338945 A263007 A104714
KEYWORD
sign,mult
AUTHOR
N. J. A. Sloane, Feb 14 2008, based on a posting to the Sequence Fans Mailing List by Yasutoshi Kohmoto, Feb 02 2005
EXTENSIONS
Extended beyond a(30) by R. J. Mathar, Aug 08 2008
More terms from Antti Karttunen, Sep 23 2017
STATUS
approved