A166261 Numbers k with property that the sum of 120 successive primes starting with prime(k) is a square.

10917, 11527, 50923, 73894, 111468, 118436, 128662, 139123, 195234, 249281, 332863, 435489, 438080, 482557, 538373, 542299, 650254, 679958, 722145, 803501, 810871, 820409, 962582, 970711, 1003544, 1027732, 1030010, 1190134, 1204929, 1305603, 1636065, 1689410

Offset: 1

Author

Zak Seidov, Oct 10 2009

Keywords

• nonn
• less

Comments

Corresponding values of s = sqrt(Sum_{i=k..k+119} prime(i)) are A166262.

Examples

			a(1) = 10917: Sum_{i=0..119} prime(10917+i) = 3734^2 = A166262(1)^2,
a(2) = 11527: Sum_{i=0..119} prime(11527+i) = 3846^2 = A166262(2)^2.
		

Crossrefs

Cf. A166262.

Cf. A064397 (2 primes), A076305 (3 primes), A072849 (4 primes), A166255 (70 primes).

Programs

Mathematica

PrimePi/@Select[Partition[Prime[Range[169*10^4]],120,1],IntegerQ[ Sqrt[ Total[ #]]]&][[All,1]] (* Harvey P. Dale, Jan 22 2019 *)

PARI

lista(nn) = {pr = primes(nn); for (i=1, nn-119, s = sum(k=i, i+119, pr[k]); if (issquare(s), print1(i, ", ")););} \\ Michel Marcus, Oct 15 2013

PARI

S=vecsum(primes(119)); p=0; q=prime(120); for(n=1,oo, issquare(S+=q-p) && print1(n","); q=nextprime(q+1); p=nextprime(p+1)) \\ It is about 25% faster to avoid "nextprime(p)" at expense of keeping the last 120 primes used in a vector p, using {my(i=Mod(0,120)); ...p[lift(i)+1]... i++}. - M. F. Hasler, Jan 04 2020

Extensions

a(30)-a(32) from Michel Marcus, Oct 15 2013

Edited by M. F. Hasler, Jan 04 2020