%I #31 Feb 03 2022 05:33:58
%S 7,18,117,239,378,843,2207,2943,4443,4662,6072,8307,8708,9872,31561,
%T 103682,271443,853932,1021693,3539232,3699356,6349657,6907607,7042807,
%U 7249325,9335094,12623932,12752043,12813848,22211431,33385282,42483057,52374157,105026693
%N Numbers m such that there exists at least one integer k < m such that m^2+1 and k^2+1 have the same prime factors.
%C For the pairs (m, k), is k always unique?
%C The pairs (m, k) are (7, 3), (18, 8), (117, 43), (239, 5), (378, 132), (843, 377), (2207, 987), (2943, 73), (4443, 53), (4662, 1568), (6072, 5118), (8307, 743), (8708, 2112), (9872, 2738), ...
%e 7 is in the sequence because of the pair (m, k) = (7, 3), 7^2+1 = 2*5^2 and 3^2+1 = 2*5 with the same prime factors 2 and 5.
%t Select[Range@ 5000, Function[m, Total@ Boole@ Table[Function[w, And[SameQ[First@ w, #], SameQ[Last@ w, #]] &@ Union@ Flatten@ w]@ Map[FactorInteger[#][[All, 1]] &, {m^2 + 1, k^2 + 1}], {k, m - 1}] > 0]] (* _Michael De Vlieger_, Feb 07 2017 *)
%o (Perl)
%o use ntheory qw(:all);
%o for (my ($m, %t) = 1 ; ; ++$m) {
%o my $k = vecprod(map{$_->[0]}factor_exp($m**2+1));
%o push @{$t{$k}}, $m;
%o if (@{$t{$k}} >= 2) {
%o print'('.join(', ',reverse(@{$t{$k}})).")\n";
%o }
%o } # _Daniel Suteu_, Feb 08 2017
%o (PARI) isok(n)=ok = 0; vn = factor(n^2+1)[,1]; for (k=1, n-1, if (factor(k^2+1)[,1] == vn, ok = 1; break);); ok; \\ _Michel Marcus_, Feb 09 2017
%o (PARI) squeeze(f)=factorback(f)\2
%o list(lim)=my(v=List(),m=Map(),t); for(n=1,lim, t=squeeze(factor(n^2+1)[,1]); if(mapisdefined(m,t), listput(v,n), mapput(m,t,0))); Vec(v) \\ _Charles R Greathouse IV_, Feb 12 2017
%Y Subsequence of A049532 (numbers n such that n^2 + 1 is not squarefree).
%Y Cf. A002522, A059591, A059592, A124809.
%K nonn
%O 1,1
%A _Michel Lagneau_, Feb 06 2017
%E a(15)-a(29) from _Daniel Suteu_, Feb 08 2017
%E a(30) from _Daniel Suteu_, Feb 10 2017
%E a(31)-a(34) from _Joerg Arndt_, Feb 11 2017