A076305 Numbers k such that prime(k) + prime(k+1) + prime(k+2) is a square.

6, 12, 59, 65, 112, 965, 1029, 1455, 1706, 1830, 1890, 2573, 3457, 4490, 4664, 5609, 7927, 9130, 10078, 10143, 12597, 18248, 19727, 20086, 20887, 21708, 22739, 25041, 26536, 28511, 29346, 29664, 29774, 33387, 39945, 40677, 46136, 49869, 58135

Offset: 1

Author

Zak Seidov, Oct 05 2002

Keywords

• nonn

Comments

See A076304 for the square roots of the sums of the three primes.

Examples

			6 is a term because prime(6) + prime(7) + prime(8) = 13 + 17 + 19 = 49 = 7^2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..600 from Harvey P. Dale)

Crossrefs

Cf. A076304 (square roots of sums), A080665 (squares = sums), A206279 (lesser of the primes).

Cf. A064397 (same for 2 primes), A072849 (4 primes), A166255 (70 primes), A166261 (120 primes).

Programs

Magma

[k:k in [1..60000]| IsSquare(&+[NthPrime(k+m):m in [0,1,2]])]; // Marius A. Burtea, Jan 04 2020

Mathematica

Select[Range[60000], IntegerQ[Sqrt[Sum[Prime[k], {k, #, # + 2}]]] &] (* Ray Chandler, Sep 26 2006 *)
Position[Partition[Prime[Range[60000]],3,1],?(IntegerQ[Sqrt[ Total[ #]]]&), 1,Heads->False]//Flatten (* _Harvey P. Dale, Sep 28 2018 *)

PARI

n=0; p=2; q=3; forprime(r=5, 1e9, n++; if(issquare(p+q+r), print1(n", ")); p=q; q=r) \\ Charles R Greathouse IV, Apr 07 2017

Formula

a(n) = A000720(A206279(n)). - M. F. Hasler, Jan 03 2020

Extensions

Corrected by Ray Chandler, Sep 26 2006