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A348594 Numbers m such that m^2 + 1 = p*q with p, q primes and m = (p + q)/2 - 1.

%I A348594 #15 Jan 26 2022 09:00:25
%S A348594 8,50,1250,1800,2450,9800,14450,20000,24200,101250,105800,135200,
%T A348594 162450,168200,204800,304200,336200,451250,480200,490050,530450,
%U A348594 696200,924800,966050,1008200,1125000,1155200,1428050,1805000,2332800,2420000,2576450,2761250,2832200
%N A348594 Numbers m such that m^2 + 1 = p*q with p, q primes and m = (p + q)/2 - 1.
%C A348594 Subsequence of A085722.
%C A348594 The corresponding pairs (p, q) of the sequence are (5, 13), (41, 61), (1201, 1301), (1741, 1861), (2381, 2521), (9661, 9941), (14281, 14621), (19801, 20201), (23981, 24421), (100801, 101701), ...
%C A348594 Property:
%C A348594 a(n) = 2* A109306(n)^2 and a(n) == 0 (mod 50) for n > 1. Proof:
%C A348594 From the relations:
%C A348594 (1)        m^2 + 1 = p*q
%C A348594 (2)        (p + q)/2 = m + 1
%C A348594 We obtain:
%C A348594 (3)        p = m + 1 - sqrt(8*m)/2
%C A348594 (4)        q = m + 1 + sqrt(8*m)/2
%C A348594 with m = 2*k^2 we obtain:
%C A348594 (5)        p = k^2 + (k-1)^2
%C A348594 (6)        q = k^2 + (k+1)^2
%C A348594 For n > 1, A109306(n) == 0 (mod 5) => 2*A109306(n)^2 == 0 (mod 50).
%e A348594 50 = 2*5^2 is in the sequence because 50^2 + 1 = 41*61 with 50 = (41 + 61)/2 - 1.
%p A348594 with(numtheory):nn:=250:printf(`%d, `,8):
%p A348594 for k from 0 to nn do:
%p A348594 n:=50*k^2:d:=factorset(n^2+1):
%p A348594   if bigomega(n^2+1)=2 and (d[1]+d[2])/2 - 1 = n
%p A348594    then
%p A348594     printf(`%d, `,n):
%p A348594     else
%p A348594   fi:
%p A348594 od:
%t A348594 q[n_] := Module[{f = FactorInteger[n^2 + 1]}, f[[;; , 2]] == {1, 1} && f[[1, 1]] + f[[2, 1]] == 2*n + 2]; Select[Range[3*10^5], q] (* _Amiram Eldar_, Jan 26 2022 *)
%o A348594 (PARI) isok(m) = my(x); if (bigomega(x=m^2+1)==2, my(f=factor(x)); (f[1,1]+f[2,1] == 2*(m+1))); \\ _Michel Marcus_, Jan 26 2022
%Y A348594 Cf. A014442, A085722, A089120, A109306, A144255.
%K A348594 nonn
%O A348594 1,1
%A A348594 _Michel Lagneau_, Jan 26 2022