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%I #23 May 31 2013 04:29:55
%S 12,18,30,42,54,60,96,102,108,120,144,150,156,174,186,210,228,252,264,
%T 270,294,312,408,420,426,456,462,510,534,540,552,558,564,570,582,588,
%U 594,600,606,654,672,696,714,774,816
%N Numbers n such that n^2 = prime(i)*prime(i+3) + prime(j)^2, for some i, j > 0, and such that prime(i+3) = prime(i) + 2*prime(j).
%C In all solutions of this equation n is divisible by 6.
%C The solution values for n = prime(i) + prime (j), when restricted by the condition prime(i+3) = prime (i) + 2*prime(j). Rather than being overly restrictive, the condition applies to the most prevalent type of solution to the equation above for n^2. See A225461 for details.
%C The equation is member of an infinite family of similar equations written as: n^2 = prime(i)*prime(i+d) + prime(j)^2, for any i,j, or d > 0. In this instance d = 3.
%C There are some additional solutions for n that do NOT obey the condition above. These are sparse but include: 60 (a 2nd time), 150, 1434, 4584 and 5190 all of which occur at low values of prime(i) and which obey the condition: n = prime(j) + 1. These are also divisible by 6, but they are excluded from the listing above.
%D A225461
%H Charles R Greathouse IV, <a href="/A225576/b225576.txt">Table of n, a(n) for n = 1..10000</a>
%e 12 is a solution value for N because 12^2 = 7*17 + 5^2 and 17 is the third prime after 7.
%o (PARI) is(n)=my(p=2,q=3,r=5,t);forprime(s=7,n+160,if(issquare(n^2-p*s,&t) && isprime(t), return(1));p=q;q=r;r=s); 0 \\ _Charles R Greathouse IV_, May 13 2013
%Y Cf. A000040.
%K nonn
%O 1,1
%A _Richard R. Forberg_, May 10 2013
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