A225576 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).

12, 18, 30, 42, 54, 60, 96, 102, 108, 120, 144, 150, 156, 174, 186, 210, 228, 252, 264, 270, 294, 312, 408, 420, 426, 456, 462, 510, 534, 540, 552, 558, 564, 570, 582, 588, 594, 600, 606, 654, 672, 696, 714, 774, 816

Offset: 1

Author

Richard R. Forberg, May 10 2013

Keywords

• nonn

Comments

In all solutions of this equation n is divisible by 6.

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.

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.

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.

Examples

			12 is a solution value for N because 12^2 = 7*17 + 5^2 and 17 is the third prime after 7.
		

References

• A225461

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040.

Programs

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