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A282092 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.

%I A282092 #31 Feb 03 2022 05:33:58
%S A282092 7,18,117,239,378,843,2207,2943,4443,4662,6072,8307,8708,9872,31561,
%T A282092 103682,271443,853932,1021693,3539232,3699356,6349657,6907607,7042807,
%U A282092 7249325,9335094,12623932,12752043,12813848,22211431,33385282,42483057,52374157,105026693
%N A282092 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 A282092 For the pairs (m, k), is k always unique?
%C A282092 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 A282092 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 A282092 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 A282092 (Perl)
%o A282092 use ntheory qw(:all);
%o A282092 for (my ($m, %t) = 1 ; ; ++$m) {
%o A282092 my $k = vecprod(map{$_->[0]}factor_exp($m**2+1));
%o A282092 push @{$t{$k}}, $m;
%o A282092 if (@{$t{$k}} >= 2) {
%o A282092 print'('.join(', ',reverse(@{$t{$k}})).")\n";
%o A282092 }
%o A282092 } # _Daniel Suteu_, Feb 08 2017
%o A282092 (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 A282092 (PARI) squeeze(f)=factorback(f)\2
%o A282092 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 A282092 Subsequence of A049532 (numbers n such that n^2 + 1 is not squarefree).
%Y A282092 Cf. A002522, A059591, A059592, A124809.
%K A282092 nonn
%O A282092 1,1
%A A282092 _Michel Lagneau_, Feb 06 2017
%E A282092 a(15)-a(29) from _Daniel Suteu_, Feb 08 2017
%E A282092 a(30) from _Daniel Suteu_, Feb 10 2017
%E A282092 a(31)-a(34) from _Joerg Arndt_, Feb 11 2017