A062703 Squares that are the sum of two consecutive primes.

36, 100, 144, 576, 1764, 2304, 3844, 5184, 7056, 8100, 12100, 14400, 14884, 30276, 41616, 43264, 48400, 53824, 57600, 69696, 93636, 106276, 112896, 138384, 148996, 166464, 168100, 197136, 206116, 207936, 219024, 220900, 224676, 272484, 298116, 302500, 352836

Offset: 1

Author

Jason Earls, Jul 11 2001

Keywords

• easy
• nonn

Examples

			prime(7) + prime(8) = 17 + 19 = 36 = 6^2.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..22054 (terms 1..100 from Harry J. Smith)

Crossrefs

Squares in A001043. See A226524 for cubes.

Cf. A074924 (square roots), A061275 (lesser of the primes), A064397 (index of that prime).

Cf. A080665 (same with sum of three consecutive primes).

Programs

Mathematica

PrevPrim[n_] := Block[{k = n - 1}, While[ !PrimeQ[k], k-- ]; k]; NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; f[n_] := Block[{m = Floor[n/2]}, s = PrevPrim[m] + NextPrim[m]; If[s == n, True, False]]; Select[ Range[550], f[ #^2] &]^2
t := Table[Prime[n] + Prime[n + 1], {n, 15000}]; Select[t, IntegerQ[Sqrt[#]] &] (* Carlos Eduardo Olivieri, Feb 25 2015 *)

PARI

{for(n=1,100,(p=precprime(n^2/2))+nextprime(p+2) == n^2 && print1(n^2", "))} \\ Zak Seidov, Feb 17 2011

PARI

A062703(n)=A074924(n)^2 \\ M. F. Hasler, Jan 03 2020

Python

from itertools import count, islice
from sympy import nextprime, prevprime
def agen(): # generator of terms
    for k in count(4, step=2):
        kk = k*k
        if prevprime(kk//2+1) + nextprime(kk//2-1) == kk:
            yield kk
print(list(islice(agen(), 37))) # Michael S. Branicky, May 24 2022

Formula

a(n) = A074924(n)^2.

a(n) = A000040(i) + A000040(i+1) with i = A064397(n) = A000720(A061275(n)). - M. F. Hasler, Jan 03 2020

Extensions

Edited by Robert G. Wilson v, Mar 02 2003

Edited (crossrefs completed, obsolete PARI code deleted) by M. F. Hasler, Jan 03 2020