This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A348889 #29 Mar 26 2022 15:07:57
%S A348889 4,5,14,22,71,140,194,218,265,602,707,724,1020,1048,1112,1642,2030,
%T A348889 2459,2695,3155,3866,4433,4756,5426,5756,8240,10046,10084,11008,15386,
%U A348889 15926,19462,21362,23092,23144,24475,35230,37634,44306,56327,64876,85352,161564,177530
%N A348889 Numbers m such that there exists at least one integer k < m where m^2 + 2 and k^2 + 2 have the same prime factors.
%C A348889 This is a subsequence of A227897 (numbers k such that k^2 + 2 is not squarefree).
%C A348889 If we consider the pairs (m, k), k is not unique, in contrast to the same problem with m^2 + 1 (see A282092) where conjecturally k seems unique.
%C A348889 The corresponding pairs (m, k) are (4, 2), (5, 1), (14, 8), (22, 2), (22, 4), (71, 11), (140, 8), (140, 14), (194, 112), (218, 40), (265, 7), (602, 146), (707, 141), ... The cases where k is not unique are given by the pairs (22, 2), (22, 4), (140, 8), (140, 14), (2695, 11), (2695, 71), (3866, 248), (3866, 2030), ...
%e A348889 4 is in the sequence because with the pair (m, k) = (4, 2), we obtain the numbers 4^2+2 = 2*3^2 and 2^2+2 = 2*3 with the same prime factors 2 and 3.
%e A348889 140 is in the sequence because with the first pair (m, k) = (140, 8), we obtain the numbers 140^2+2 = 2*3*11^2 and 8^2+2 = 2*3*11 with the same prime factors 2, 3 and 11; with the second pair (m, k) = (140, 14), we obtain the numbers 140^2+2 = 2*3*11^2 and 14^2+2 = 2*3^2*11 with the same prime factors 2, 3 and 11.
%t A348889 Select[Range@ 2000, Function[m, Total@ Boole@ Table[Function[w, And[SameQ[First@ w, #], SameQ[Last@ w, #]] &@ Union@ Flatten@ w]@ Map[FactorInteger[#][[All, 1]] &, {m^2 + 2, k^2 + 2}], {k, m - 1}] > 0]] (* program from _Michael De Vlieger_, adapted for the sequence - see A282092 *)
%o A348889 (Magma) M:=178000; S:=[]; for k in [1..M] do S[k]:=[&*PrimeDivisors(k^2+2),k]; end for; S:=Sort(S); a:=[]; for j in [2..#S] do if S[j][1] eq S[j-1][1] then a[#a+1]:=S[j][2]; end if; end for; a:=Sort(a); a; // _Jon E. Schoenfield_, Jan 28 2022
%o A348889 (PARI) isok(m) = {if (!issquarefree(m^2+2), my(f=factor(m^2+2)[,1]); for (k=1, m-1, if (factor(k^2+2)[, 1] == f, return(1));););} \\ _Michel Marcus_, Feb 02 2022
%Y A348889 Cf. A227897, A282092.
%K A348889 nonn
%O A348889 1,1
%A A348889 _Michel Lagneau_, Jan 28 2022
%E A348889 More terms from _Jinyuan Wang_, Jan 28 2022