A272349 -id:A272349 - cp's OEIS frontend

A227470 Least k such that n divides sigma(n*k).

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

Offset: 1

Alex Ratushnyak, Jul 12 2013

nonn

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

			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.

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

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

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

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

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

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

A097018 a(n) is the least number such that sigma(a(n)) is divisible by the n-th prime.

Original entry on oeis.org

3, 2, 8, 4, 43, 9, 67, 37, 137, 173, 16, 73, 163, 257, 281, 211, 353, 169, 401, 283, 256, 157, 331, 1024, 193, 1009, 617, 641, 653, 677, 64, 523, 547, 277, 1489, 1811, 313, 977, 1669, 691, 1789, 1447, 4201, 1543, 787, 397, 421, 1783, 907, 457, 3727, 1433, 3373

Offset: 1

Labos Elemer, Aug 23 2004

nonn

Note that a(n) always exists, because sigma(2^(p-2)) = 2^(p-1)-1 is divisible by p for p>2 (by Fermat's little theorem), so there is always a candidate for a(n). Compare A227470, A272349. - N. J. A. Sloane, May 01 2016

			n=11: a(11)=16 is the least number x such that sigma(x) is divisible by the 11th prime, 31.

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A000203, A000040, A272349, A227470.

sol:=[]; p:=PrimesUpTo(10000); for n in [1..53] do k:=2; while Max(PrimeDivisors(SumOfDivisors(k))) ne p[n] do k:=k+1; end while; sol[n]:=k; end for; sol; // Marius A. Burtea, Jun 05 2019

ln[n_]:=Module[{x=1,p=Prime[n]},While[!Divisible[DivisorSigma[ 1,x], p],x++];x]; Array[ln,60] (* Harvey P. Dale, Sep 07 2014 *)
Module[{nn=5000,ds},ds=DivisorSigma[1,Range[nn]];Table[Position[ds,?(Divisible[#,n]&),1,1],{n,Prime[Range[60]]}]]//Flatten (* Much faster than the first program *) (* _Harvey P. Dale, May 18 2018 *)

sigma_hunt(x)=local(n=0,g);while(n++,g=sigma(n);if(g%x,,return(n)));
for(x=1,50,print1(sigma_hunt(prime(x))", ")) /* Phil Carmody, Mar 01 2013 */

cp's OEIS Frontend

A227470 Least k such that n divides sigma(n*k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula