A074208 Least k > 1 such that n divides sigma(k) - k.

2, 6, 4, 9, 14, 6, 8, 10, 15, 14, 20, 24, 27, 22, 16, 12, 39, 24, 48, 34, 18, 20, 52, 90, 40, 46, 42, 28, 68, 78, 32, 56, 45, 62, 84, 24, 70, 48, 66, 44, 63, 30, 50, 82, 78, 52, 116, 90, 75, 40, 132, 96, 80, 42, 36, 106, 99, 68, 148, 120, 130, 118, 64, 56, 117, 54, 136, 112

Offset: 1

Author

Benoit Cloitre, Sep 17 2002

Keywords

• nonn
• easy

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A070982.

Programs

Mathematica

a = ConstantArray[1, 68]; k = 1; While[Length[vac = Flatten[Position[a, 1]]] > 0, k++; a[[Intersection[Divisors[DivisorSigma[1, k] - k], vac]]] *= k]; a (* Ivan Neretin, May 15 2015 *)
lk[n_]:=Module[{k=2},While[!Divisible[DivisorSigma[1,k]-k,n],k++];k]; Array[lk,70] (* Harvey P. Dale, Oct 23 2017 *)

PARI

a(n)=if(n<0,0,s=2; while((sigma(s)-s)%n>0,s++); s)

PARI

first(n)=my(res = vector(n), todo = n - 1, k = 2); res[1] = 2; while(todo > 0, d = divisors(sigma(k) - k); for(i=2, #d, if(d[i] <= n && res[d[i]] == 0, res[d[i]] = k; todo--)); k++); res \\ David A. Corneth, Oct 23 2017

Formula

Conjecture: Sum_{k=1..n} a(k) = O(n^{2+epsilon}) for any epsilon > 0.

Between n = 90000 and 100000, Sum_{k=1..n} a(k)/n^2 slowly but not monotonically increases from 1.0007 to 1.0023. At n = 10^6, it's about 1.0147. - David A. Corneth, Oct 23 2017