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%I #20 Jan 03 2020 23:23:37
%S 7,11,29,31,43,151,157,191,209,217,221,263,311,359,367,407,493,533,
%T 563,565,637,781,815,823,841,859,881,929,959,997,1013,1019,1021,1087,
%U 1199,1211,1297,1353,1471,1573,1613,1683,1685,1733,1735,1739,1751,1761,1769
%N Numbers n such that n^2 is a sum of three successive primes.
%H Zak Seidov, <a href="/A076304/b076304.txt">Table of n, a(n) for n = 1..255</a>
%F a(n) = sqrt(prime(i) + prime(i+1) + prime(i+2)) where i = A076305(n). [Corrected by _M. F. Hasler_, Jan 03 2020]
%e 7 is in this sequence because 7^2 = 49 = p(6) + p(7) + p(8) = 13 + 17 + 19.
%t Select[Table[Sqrt[Sum[Prime[k], {k, n, n + 2}]], {n, 100000}], IntegerQ] (* _Ray Chandler_, Sep 29 2006 *)
%t Select[Sqrt[#]&/@(Total/@Partition[Prime[Range[90000]],3,1]),IntegerQ] (* _Harvey P. Dale_, Feb 23 2011 *)
%o (PARI) is(n, p=precprime(n^2/3), q=nextprime(p+1), t=n^2-p-q)=isprime(t) && t==if(t>q,nextprime(q+1),precprime(p-1)) \\ _Charles R Greathouse IV_, May 26 2013; edited by _M. F. Hasler_, Jan 03 2020
%o (PARI) A76304=[7]; apply( A076304(n)={if(n>#A76304, my(i=#A76304, N=A76304[i]); A76304=concat(A76304, vector(n-i,i, until( is(N+=2),);N))); A76304[n]}, [1..99]) \\ _M. F. Hasler_, Jan 03 2020
%Y Cf. A206279 (smallest of the 3 primes), A076305 (index of that prime), A080665 (squares = sums), A122560 (subsequence of primes).
%K nonn,easy
%O 1,1
%A _Zak Seidov_, Oct 05 2002
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