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A227470 Least k such that n divides sigma(n*k). 5
1, 3, 2, 3, 8, 1, 4, 7, 10, 4, 43, 2, 9, 2, 8, 21, 67, 5, 37, 6, 20, 43, 137, 5, 149, 9, 34, 1, 173, 4, 16, 21, 27, 64, 76, 22, 73, 37, 6, 3, 163, 10, 257, 43, 6, 137, 281, 11, 52, 76, 67, 45, 211, 17, 109, 4, 49, 173, 353, 2, 169, 8, 32, 93, 72, 27, 401, 67 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Theorem: a(n) always exists.

Proof: If n is a power of a prime, say n = p^a, then, by Euler's generalization of Fermat's little theorem and the multiplicative property of sigma, one can take k = x^(p^a-p^(a-1)-1) where x is a different prime from p. Similarly. if n = p^a*q^b, then take k = x^(p^a-p^(a-1)-1)*y^(q^b-q^(b-1)-1) where {x,y} are primes different from {p,q}. And so on. These k's have the desired property, and so there is always at least one candidate for the minimal k. - N. J. A. Sloane, May 01 2016

LINKS

R. J. Mathar, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = A272349(n)/n. - R. J. Mathar, May 06 2016

EXAMPLE

Least k such that 9 divides sigma(9*k) is k = 10: sigma(90) = 234 = 9*26. So a(9) = 10.

Least k such that 89 divides sigma(89*k) is k = 1024: sigma(89*1024) = 184230 = 89*2070. So a(89) = 1024.

MAPLE

A227470 := proc(n)

    local k;

    for k from 1 do

        if modp(numtheory[sigma](k*n), n) =0 then

            return k;

        end if;

    end do:

end proc: # R. J. Mathar, May 06 2016

MATHEMATICA

lknds[n_]:=Module[{k=1}, While[!Divisible[DivisorSigma[1, k*n], n], k++]; k]; Array[lknds, 70] (* Harvey P. Dale, Jul 10 2014 *)

PROG

(PARI) a227470(n) = {k=1; while(sigma(n*k)%n != 0, k++); k} \\ Michael B. Porter, Jul 15 2013

CROSSREFS

Indices of 1's: A007691.

Cf. A000203, A227302, A227303, A097018.

See A272349 for the sequence [n*a(n)]. - N. J. A. Sloane, May 01 2016

Sequence in context: A225695 A226469 A073341 * A218396 A331926 A070982

Adjacent sequences:  A227467 A227468 A227469 * A227471 A227472 A227473

KEYWORD

nonn

AUTHOR

Alex Ratushnyak, Jul 12 2013

STATUS

approved

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Last modified February 18 09:15 EST 2020. Contains 332011 sequences. (Running on oeis4.)