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

1,1

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.

Links

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

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A005574, A085722.

Sequence in context: A370687 A003086 A102958 * A220197 A299880 A358928

Adjacent sequences: A321792 A321793 A321794 * A321796 A321797 A321798

Keyword

nonn

Author

Michel Lagneau, Nov 19 2018

Status

approved