A321795 Numbers m such that m^2+1 is prime with (m-1)^2+1 and (m+1)^2+1 semiprimes.

4, 10, 170, 570, 780, 950, 1420, 2380, 2730, 3850, 4120, 4300, 5850, 6360, 6460, 6800, 6970, 7100, 7240, 8720, 9630, 10150, 10580, 11010, 11170, 11830, 12300, 14290, 16330, 17670, 17810, 17850, 17860, 18940, 19030, 20500, 21930, 23960, 24490, 25830, 26050

Offset: 1

Author

Michel Lagneau, Nov 19 2018

Keywords

• nonn

Comments

Subsequence of A005574.

For n>1, a(n) == 0 (mod 10).

The corresponding pairs of semiprimes ((m-1)^2+1, (m+1)^2+1) are of the form (2p, 2q) with p, q primes == 1 (mod 10). So, a(n) = (q - p)/2 and a(n)^2 + 1 = p + q - 1.

Examples

			10 is in the sequence because 10^2 + 1 = 101 is prime, and 9^2 + 1 = 2*41, 11^2 + 1 = 2*61 are semiprimes.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A005574, A085722.

Programs

Mathematica

Select[Range[50000],PrimeOmega[(#-1)^2+1]==2&&PrimeQ[#^2+1]&&PrimeOmega[(#+1)^2+1]==2&]
Mean/@SequencePosition[Table[Which[PrimeQ[m^2+1],1,PrimeOmega[m^2+1]==2,2,True,0],{m,30000}],{2,1,2}] (* Requires Mathematica version 10 or later *)  (* Harvey P. Dale, Sep 04 2019 *)

PARI

isok(m) = isprime(m^2+1) && (bigomega((m-1)^2+1) == 2) && (bigomega((m+1)^2+1) == 2); \\ Michel Marcus, Nov 20 2018