%I #7 Dec 08 2018 00:34:44

%S 3,5,15,25,205,715,1095,1315,1615,2055,2405,2925,3755,4615,4795,5015,

%T 5055,5475,6785,7855,8115,8175,9425,9475,10415,10845,11025,11245,

%U 12335,12765,15225,16225,16395,16405,18145,18175,18275,21345,21915,22905,23165,23815

%N Numbers m such that m^2+1 is semiprime with (m-1)^2+1 and (m+1)^2+1 primes.

%C Subsequence of A085722.

%C For n>1, a(n) == 5 (mod 10).

%C The corresponding pairs of primes (p, q) = ((m-1)^2+1, (m+1)^2+1) are congruent to 7 (mod 10), and the semiprimes are of the form m^2+1 = 2r where r is congruent to 3 (mod 10). So, a(n) = (q - 2r - 1)/2 = (2r - p + 1)/2 = (q - p)/4.

%e 15 is in the sequence because 15^2 + 1 = 2*113 is semiprime, and 14^2 + 1 = 197, 16^2 + 1 = 257 are prime numbers.

%t Select[Range[50000],PrimeQ[(#-1)^2+1]&&PrimeOmega [#^2+1]==2&&PrimeQ[(#+1)^2+1]&]

%o (PARI) isok(m) = (bigomega(m^2+1) == 2) && isprime((m-1)^2+1) && isprime((m+1)^2+1); \\ _Michel Marcus_, Nov 23 2018

%Y Cf. A005574, A085722, A321795.

%K nonn

%O 1,1

%A _Michel Lagneau_, Nov 23 2018