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

1,1

Links

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

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

Example

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

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 *)

Prog

(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

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).

Sequence in context: A069057 A342402 A348826 * A043438 A044223 A044604

Adjacent sequences: A062700 A062701 A062702 * A062704 A062705 A062706

Keyword

easy,nonn

Author

Jason Earls, Jul 11 2001

Extensions

Edited by Robert G. Wilson v, Mar 02 2003

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

Status

approved