VOOZH about

URL: https://oeis.org/A074924

⇱ A074924 - OEIS

login
A074924
Numbers whose square is the sum of two successive primes.
22
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.
MAPLE
filter:= proc(n) local t; t:= n^2/2; prevprime(ceil(t)) + nextprime(floor(t)) = n^2 end proc:
select(filter, [$3..1000]); # Robert Israel, Nov 19 2024
MATHEMATICA
Select[Sqrt[#]&/@(Total/@Partition[Prime[Range[50000]], 2, 1]), IntegerQ] (* Harvey P. Dale, Oct 04 2014 *)
f@n_ := Sqrt@Select[(2*Range@n)^2, # == Plus @@ NextPrime[#/2, {-1, 1}] &]; f@485 (* Hans Rudolf Widmer, Nov 19 2024 *)
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
KEYWORD
nonn
AUTHOR
Zak Seidov, Oct 02 2002
EXTENSIONS
Crossrefs section corrected and extended by M. F. Hasler, Jan 03 2020
STATUS
approved