A227303 -id:A227303 - 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

A245774 Numbers k that divide 3*sigma(k).

Original entry on oeis.org

1, 3, 6, 12, 28, 84, 120, 234, 270, 496, 672, 1080, 1488, 1638, 6048, 6552, 8128, 24384, 30240, 32760, 35640, 199584, 435708, 523776, 2142720, 2178540, 4713984, 12999168, 18506880, 23569920, 33550336, 36197280, 45532800

Offset: 1

Jaroslav Krizek, Aug 26 2014

nonn

Numbers k that divide 3*A000203(k).
Supersequence of A007691 and A245775.
Union of A007691 and 3*A227303. - Robert Israel, Aug 26 2014

			Number 12 is in the sequence because 12 divides 3*sigma(12) = 3*28.

Cf. A000203 (sum of divisors), A007691 (multiply-perfect numbers).
Cf. A227303 (n divides sigma(3n)), A245775 (denominator(sigma(n)/n) = 3).
Cf. A272027 (3*sigma(n)).

[n: n in [1..3000000] | Denominator(3*(SumOfDivisors(n))/n) eq 1]

select(n -> 3*numtheory:-sigma(n) mod n = 0, [$1..10^6]); # Robert Israel, Aug 26 2014

a245774[n_Integer] := Select[Range[n], Divisible[3*DivisorSigma[1, #], #] == True &]; a245774[10^7] (* Michael De Vlieger, Aug 27 2014 *)

for(n=1,10^9,if((3*sigma(n))%n==0,print1(n,", "))) \\ Derek Orr, Aug 26 2014

A341623 Numbers k such that sigma(3k) = 8k.

Original entry on oeis.org

28, 90, 496, 546, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176

Offset: 1

Antti Karttunen, Feb 19 2021

Every perfect number P greater than 6 (so, P is not divisible by 3) will be found in this sequence. Proof: sigma(3*P) = sigma(3)*sigma(P) = 4*(2*P) = 8*P. - Timothy L. Tiffin, Aug 26 2021

Solutions are integers y/3 where sigma(y)/y = 8/3. - Michel Marcus, Aug 27 2021

			546 is a term, since sigma(3*546) = sigma(1638) = 4368 = 8*546. - _Timothy L. Tiffin_, Aug 26 2021

Cf. A000396 (subsequence, apart from its terms that are divisible by 3).
Subsequence of A005101 and A227303.
Cf. also A000203, A144613, A326051, A336702, A347261.

Select[Range[5*10^9], DivisorSigma[1, 3*#] == 8*# &] (* Timothy L. Tiffin, Aug 26 2021 *)
Do[If[DivisorSigma[1, 3*k] == 8*k, Print[k]], {k, 5*10^9}] (* Timothy L. Tiffin, Aug 26 2021 *)

a(7)-a(8) from Martin Ehrenstein, Mar 06 2021

a(9)-a(10) from Michel Marcus, Aug 27 2021

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

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A245774 Numbers k that divide 3*sigma(k).

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A341623 Numbers k such that sigma(3k) = 8k.

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cp's OEIS Frontend

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

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A245774 Numbers k that divide 3*sigma(k).

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Magma

Maple

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A341623 Numbers k such that sigma(3*k) = 8*k.

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A341623 Numbers k such that sigma(3k) = 8k.