A321985 Numbers m such that m^2+1 is semiprime with (m-1)^2+1 and (m+1)^2+1 primes.

3, 5, 15, 25, 205, 715, 1095, 1315, 1615, 2055, 2405, 2925, 3755, 4615, 4795, 5015, 5055, 5475, 6785, 7855, 8115, 8175, 9425, 9475, 10415, 10845, 11025, 11245, 12335, 12765, 15225, 16225, 16395, 16405, 18145, 18175, 18275, 21345, 21915, 22905, 23165, 23815

Offset

1,1

Comments

Subsequence of A085722.

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

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.

Links

Table of n, a(n) for n=1..42.

Example

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.

Mathematica

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

Prog

(PARI) isok(m) = (bigomega(m^2+1) == 2) && isprime((m-1)^2+1) && isprime((m+1)^2+1); \\ Michel Marcus, Nov 23 2018

Crossrefs

Cf. A005574, A085722, A321795.

Sequence in context: A056487 A146582 A349967 * A301524 A202212 A253790

Adjacent sequences: A321982 A321983 A321984 * A321986 A321987 A321988

Keyword

nonn

Author

Michel Lagneau, Nov 23 2018

Status

approved