A070155 Numbers k such that k-1, k+1 and k^2+1 are prime numbers.

4, 6, 150, 180, 240, 270, 420, 570, 1290, 1320, 2310, 2550, 2730, 3360, 3390, 4260, 4650, 5850, 5880, 6360, 6780, 9000, 9240, 9630, 10530, 10890, 11970, 13680, 13830, 14010, 14550, 16230, 16650, 18060, 18120, 18540, 19140, 19380, 21600, 21840

Offset: 1

Author

Labos Elemer, Apr 23 2002

Keywords

• easy
• nonn

Comments

Essentially the same as A129293. - R. J. Mathar, Jun 14 2008

Solutions to the equation: A000005(n^4-1) = 8. - Enrique Pérez Herrero, May 03 2012

Terms > 6 are multiples of 30. Subsequence of A070689. - Zak Seidov, Nov 12 2012

{a(n)-1} is a subsequence of A157468; for n>1, {a(n)^2+2} is a subsequence of A242720. - Vladimir Shevelev, Aug 31 2014

Examples

			150 is a term since 149, 151 and 22501 are all primes.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Crossrefs

Cf. A000005, A000720, A001359, A006512, A014574, A002496, A005574, A070156, A070689, A129293, A157468, A242720.

Programs

Maple

select(n -> isprime(n-1) and isprime(n+1) and isprime(n^2+1), [seq(2*i,i=1..10000)]); # Robert Israel, Sep 02 2014

Mathematica

Do[s=n; If[PrimeQ[s-1]&&PrimeQ[s+1]&&PrimeQ[1+s^2], Print[n]], {n, 1, 1000000}]
Select[Range[22000],AllTrue[{#+1,#-1,#^2+1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 19 2014 *)

PARI

is(k) = isprime(k-1) && isprime(k+1) && isprime(k^2+1); \\ Amiram Eldar, Apr 15 2024

Formula

For n>1, a(n)^2 = A242720(pi(a(n)-2)) - 2, where pi(n) is the prime counting function (A000720). - Vladimir Shevelev, Sep 02 2014