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A062703
Squares that are the sum of two consecutive primes.
15
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
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