A126690 Multiplicative function defined for prime powers by a(p^k) = p + p^2 + p^3 + ... + p^(k-1) - 1 (k >= 1).

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), A046090.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

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

Crossrefs

Sequence in context: A326327 A113103 A033325 * A338945 A263007 A104714

Adjacent sequences: A126687 A126688 A126689 * A126691 A126692 A126693

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