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
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