A321328 a(n) is the smallest number k such that k = (sigma(n*(sigma(k)-k)) - n*(sigma(k)-k))/n.

6, 20, 14, 4, 10, 26, 1012, 8, 1442, 68, 376, 38, 1660, 14, 506, 574, 352, 117, 590, 22, 254, 1292, 460, 82, 26108, 416, 266, 10, 3496, 15, 124, 32, 470, 5176, 658, 362, 104696, 152, 19305, 51, 12782, 62, 618770, 232, 15561, 1136, 4136, 1006, 8588, 49166, 154, 13988

Offset: 1

Author

Paolo P. Lava, Nov 05 2018

Keywords

• nonn

Comments

A sort of generalization of amicable numbers where x = n*(sigma(k)-k), y = (sigma(x)-x)/n = k and x >= y.

All the numbers that satisfy the equation for n=1 are listed in A206708.

a(n) = n for n = 4, 8, 14, 32, 128, 2366, 8193, 131072, etc.

In particular a(n) = n if n = 2^p where p is a Mersenne exponent (A000043).

Examples

			a(7) = 1012 because (sigma(7*(sigma(1012)-1012)) - 7*(sigma(1012)-1012))/7 = (sigma(7*1004) - 7*1004)/7 = (14112-7028)/7 = 7084/7 = 1012 and this is the least number to have this property.
		

Links

• Paolo P. Lava, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A000040, A000043, A000203, A206708.

Programs

Maple

with(numtheory): P:=proc(q) local k,n; for n from 1 to q do
for k from 1 to q do if (sigma(n*(sigma(k)-k))-n*(sigma(k)-k))/n=k
then print(k); break; fi; od; od; end: P(10^6);

Mathematica

s[n_] := DivisorSigma[1,n]-n; a[n_] := Module[{k=2}, While[k != s[n*s[k]]/n, k++];k]; Array[a, 52] (* Amiram Eldar, Nov 06 2018 *)

PARI

f(n,k) = {my(sk = sigma(k)-k); iferr((sigma(n*sk)-n*sk)/n, E, 0);}
a(n) = {my(k=1); while (k != f(n,k), k++); k;} \\ Michel Marcus, Nov 06 2018