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%I #9 Feb 13 2016 23:10:36
%S 901,10001,20737,75077,234257,266257,276677,571537,1094117,1562501,
%T 2937797,3261637,3363557,5216657,5953601,6812101,8643601,12418577,
%U 14622977,17556101,25847057,33016517,45778757,56040197,94984517,98406401,106296101,169624577,174504101
%N Composite numbers m such that for any positive integers a < b, if a * b = m then b - a is a perfect square.
%C It appears that a(n) is semiprime => this sequence is included in A143416.
%C The sequence is probably infinite.
%C Property:
%C a(n) == 1 (mod 4), a(n)== 1 or 5 (mod 6), a(n)== 1 or 7 (mod 10), a(n)== 1 or 5 (mod 12), a(n) == 1 or 5 (mod 16), a(n)== 1 or 17 (mod 20), a(n)== 1, 5 or 17 (mod 32).
%C We find multiplicative groups (mod q) with q = 6, 12, 24.
%C Example with q = 24:
%C a(n) == {1, 5, 13, 17} mod 24 => the set {1, 5, 13, 17} is a multiplicative group (mod 24):
%C 5^2 == 1 mod 24;
%C 13^2 == 1 mod 24;
%C 17^2 == 1 mod 24;
%C 5*13 == 17 mod 24;
%C 5*17 == 13 mod 24;
%C 13*17 == 5 mod 24.
%e 901 is in the sequence because 901 = 1*901 = 17*53 => 901-1 = 30^2 and 53 - 17 = 6^2.
%t Do[ds=Divisors[n];If[EvenQ[Length[ds]],ok=True;k=1;While[k<=Length[ds]/2&&(ok=IntegerQ[Sqrt[Abs[ds[[k]]-ds[[-k]]]]]&&!PrimeQ[n]),k++];If[ok,Print[n]]],{n,2,10^5}]
%Y Cf. A143416.
%K nonn
%O 1,1
%A _Michel Lagneau_, Feb 07 2016
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