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

%I A321328 #32 Dec 08 2018 17:09:07
%S A321328 6,20,14,4,10,26,1012,8,1442,68,376,38,1660,14,506,574,352,117,590,22,
%T A321328 254,1292,460,82,26108,416,266,10,3496,15,124,32,470,5176,658,362,
%U A321328 104696,152,19305,51,12782,62,618770,232,15561,1136,4136,1006,8588,49166,154,13988
%N A321328 a(n) is the smallest number k such that k = (sigma(n*(sigma(k)-k)) - n*(sigma(k)-k))/n.
%C A321328 A sort of generalization of amicable numbers where x = n*(sigma(k)-k), y = (sigma(x)-x)/n = k and x >= y.
%C A321328 All the numbers that satisfy the equation for n=1 are listed in A206708.
%C A321328 a(n) = n for n = 4, 8, 14, 32, 128, 2366, 8193, 131072, etc.
%C A321328 In particular a(n) = n if n = 2^p where p is a Mersenne exponent (A000043).
%H A321328 Paolo P. Lava, <a href="/A321328/b321328.txt">Table of n, a(n) for n = 1..200</a>
%e A321328 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.
%p A321328 with(numtheory): P:=proc(q) local k,n; for n from 1 to q do
%p A321328 for k from 1 to q do if (sigma(n*(sigma(k)-k))-n*(sigma(k)-k))/n=k
%p A321328 then print(k); break; fi; od; od; end: P(10^6);
%t A321328 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 *)
%o A321328 (PARI) f(n,k) = {my(sk = sigma(k)-k); iferr((sigma(n*sk)-n*sk)/n, E, 0);}
%o A321328 a(n) = {my(k=1); while (k != f(n,k), k++); k;} \\ _Michel Marcus_, Nov 06 2018
%Y A321328 Cf. A000040, A000043, A000203, A206708.
%K A321328 nonn
%O A321328 1,1
%A A321328 _Paolo P. Lava_, Nov 05 2018