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

1,1

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

Links

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

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

Example

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A355232 A176493 A355230 * A280778 A197882 A074124

Adjacent sequences: A070152 A070153 A070154 * A070156 A070157 A070158

Keyword

easy,nonn,changed

Author

Labos Elemer, Apr 23 2002

Status

approved