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

%I A074924 #51 Dec 03 2024 12:45:03
%S A074924 6,10,12,24,42,48,62,72,84,90,110,120,122,174,204,208,220,232,240,264,
%T A074924 306,326,336,372,386,408,410,444,454,456,468,470,474,522,546,550,594,
%U A074924 600,630,640,642,686,740,750,762,766,788,802,852,876,882,920,936,970
%N A074924 Numbers whose square is the sum of two successive primes.
%H A074924 Zak Seidov, <a href="/A074924/b074924.txt">Table of n, a(n) for n = 1..22054</a> (all terms up to 10^6).
%F A074924 a(n) = sqrt(A062703(n)). - _Zak Seidov_, May 26 2013
%F A074924 a(n) = A000040(i) + A000040(i+1) with i = A064397(n) = A000720(A061275(n)). - _M. F. Hasler_, Jan 03 2020
%e A074924 6^2 = 17 + 19, 1610^2 = 1296041 + 1296059.
%p A074924 filter:= proc(n) local t; t:= n^2/2; prevprime(ceil(t)) + nextprime(floor(t)) = n^2 end proc:
%p A074924 select(filter, [$3..1000]); # _Robert Israel_, Nov 19 2024
%t A074924 Select[Sqrt[#]&/@(Total/@Partition[Prime[Range[50000]],2,1]),IntegerQ] (* _Harvey P. Dale_, Oct 04 2014 *)
%t A074924 f@n_ := Sqrt@Select[(2*Range@n)^2, # == Plus @@ NextPrime[#/2, {-1, 1}] &]; f@485 (* _Hans Rudolf Widmer_, Nov 19 2024 *)
%o A074924 (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
%o A074924 (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
%o A074924 (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
%o A074924 (Python)
%o A074924 from itertools import count, islice
%o A074924 from sympy import nextprime, prevprime
%o A074924 def agen(): # generator of terms
%o A074924     for k in count(4, step=2):
%o A074924         kk = k*k
%o A074924         if prevprime(kk//2+1) + nextprime(kk//2-1) == kk:
%o A074924             yield k
%o A074924 print(list(islice(agen(), 54))) # _Michael S. Branicky_, May 24 2022
%Y A074924 Square roots of squares in A001043.
%Y A074924 Cf. A062703 (the squares), A061275 (lesser of the primes), A064397 (index of that prime).
%Y A074924 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).
%K A074924 nonn
%O A074924 1,1
%A A074924 _Zak Seidov_, Oct 02 2002
%E A074924 Crossrefs section corrected and extended by _M. F. Hasler_, Jan 03 2020