A238222 Numbers m with property that m^2 + (m+1)^2 and (m+1)^2 + (m+2)^2 are prime.

1, 4, 24, 29, 34, 69, 84, 99, 109, 224, 229, 259, 284, 289, 319, 389, 409, 474, 489, 494, 514, 589, 679, 694, 709, 749, 759, 844, 949, 1079, 1099, 1134, 1174, 1189, 1194, 1269, 1294, 1304, 1329, 1364, 1409, 1474, 1714, 1749, 1784, 1844, 1854, 1924, 2014, 2059, 2099, 2149

Offset: 1

Author

Zak Seidov, Feb 21 2014

Keywords

• nonn

Comments

Integers m such both m and m+1 are terms in A027861.

All corresponding primes are == 1 mod 4 (A002144 Pythagorean primes) and terms in A027862.

No such m such that also (m+2)^2 + (m+3)^2 is prime.

Examples

			1 is in the sequence because 1^2+2^2 = 5 and 2^2+3^2 = 13 are both prime.
4 is in the sequence because 4^2+5^2 = 41 and 5^2+6^2 = 61 are both prime.
		

Links

• Zak Seidov, Table of n, a(n) for n = 1..2000

Crossrefs

Cf. A002144, A062067, A027862. Subsequence of A027861.

Programs

Mathematica

Reap[Do[If[PrimeQ[k^2+(k+1)^2]&&PrimeQ[(k+1)^2+(k+2)^2],Sow[k]],{k,2000}]][[2,1]]
Select[Range[2500],AllTrue[{#^2+(#+1)^2,(#+1)^2+(#+2)^2},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 25 2017 *)

PARI

s=[]; for(m=1, 2500, if(isprime(m^2+(m+1)^2) && isprime((m+1)^2+(m+2)^2), s=concat(s, m))); s \\ Colin Barker, Feb 21 2014