

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.


0



4, 5, 14, 22, 71, 140, 194, 218, 265, 602, 707, 724, 1020, 1048, 1112, 1642, 2030, 2459, 2695, 3155, 3866, 4433, 4756, 5426, 5756, 8240, 10046, 10084, 11008, 15386, 15926, 19462, 21362, 23092, 23144, 24475, 35230, 37634, 44306, 56327, 64876, 85352, 161564, 177530
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OFFSET

1,1


COMMENTS

This is a subsequence of A227897 (numbers k such that k^2 + 2 is not squarefree).
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.
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), ...


LINKS

Table of n, a(n) for n=1..44.


EXAMPLE

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.
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.


MATHEMATICA

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 *)


PROG

(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[j1][1] then a[#a+1]:=S[j][2]; end if; end for; a:=Sort(a); a; // Jon E. Schoenfield, Jan 28 2022
(PARI) isok(m) = {if (!issquarefree(m^2+2), my(f=factor(m^2+2)[, 1]); for (k=1, m1, if (factor(k^2+2)[, 1] == f, return(1)); ); ); } \\ Michel Marcus, Feb 02 2022


CROSSREFS

Cf. A227897, A282092.
Sequence in context: A191790 A246984 A306897 * A006904 A200177 A241863
Adjacent sequences: A348886 A348887 A348888 * A348890 A348891 A348892


KEYWORD

nonn


AUTHOR

Michel Lagneau, Jan 28 2022


EXTENSIONS

More terms from Jinyuan Wang, Jan 28 2022


STATUS

approved



