OFFSET
1,1
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).
LINKS
Paolo P. Lava, Table of n, a(n) for n = 1..200
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Paolo P. Lava, Nov 05 2018
STATUS
approved