A074924 Numbers whose square is the sum of two successive primes.

6, 10, 12, 24, 42, 48, 62, 72, 84, 90, 110, 120, 122, 174, 204, 208, 220, 232, 240, 264, 306, 326, 336, 372, 386, 408, 410, 444, 454, 456, 468, 470, 474, 522, 546, 550, 594, 600, 630, 640, 642, 686, 740, 750, 762, 766, 788, 802, 852, 876, 882, 920, 936, 970

Offset

1,1

Links

Zak Seidov, Table of n, a(n) for n = 1..22054 (all terms up to 10^6).

Formula

a(n) = sqrt(A062703(n)). - Zak Seidov, May 26 2013

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

Example

6^2 = 17 + 19, 1610^2 = 1296041 + 1296059.

Mathematica

Select[Sqrt[#]&/@(Total/@Partition[Prime[Range[50000]], 2, 1]), IntegerQ] (* Harvey P. Dale, Oct 04 2014 *)

Prog

(PARI) is(n)=if(n%2, return(0)); nextprime(n^2/2+1)+precprime(n^2/2)==n^2 \\ Charles R Greathouse IV, Apr 29 2015
(PARI) select( {is_A074924(n)=!bittest(n=n^2, 0) && precprime(n\2)+nextprime(n\/2)==n}, [1..999]) \\ M. F. Hasler, Jan 03 2020
(PARI) A74924=[6]; apply( A074924(n)={while(n>#A74924, my(N=A74924[#A74924]); until( is_A074924(N+=2), ); A74924=concat(A74924, N)); A74924[n]}, [1..99]) \\ 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 k
print(list(islice(agen(), 54))) # Michael S. Branicky, May 24 2022

Crossrefs

Square roots of squares in A001043.

Cf. A062703 (the squares), A061275 (lesser of the primes), A064397 (index of that prime).

Cf. A064397 (numbers n such that prime(n) + prime(n+1) is a square), A071220 (prime(n) + prime(n+1) is a cube), A074925 (n^3 is sum of 2 consecutive primes).

Sequence in context: A101086 A342648 A297620 * A064166 A373678 A107371

Adjacent sequences: A074921 A074922 A074923 * A074925 A074926 A074927

Keyword

nonn

Author

Zak Seidov, Oct 02 2002

Extensions

Crossrefs section corrected and extended by M. F. Hasler, Jan 03 2020

Status

approved