A218793 Numbers that can be written as p^2 + 3pq + q^2 with prime p and q.

20, 31, 45, 59, 79, 95, 121, 125, 179, 191, 229, 245, 251, 295, 311, 389, 395, 401, 451, 479, 491, 541, 569, 605, 671, 695, 719, 745, 809, 845, 899, 971, 1019, 1061, 1109, 1111, 1121, 1151, 1249, 1271, 1301, 1409, 1445, 1451, 1499, 1595, 1619, 1661, 1711

Offset

1,1

Comments

Sequence A218771 is the subsequence of primes in this sequence.

Links

M. F. Hasler, Table of n, a(n) for n = 1..7196 (all terms below 10^6).

Example

a(1) = 20 = p^2+3pq+q^2 for p=q=2, in the same way all numbers of the form 5p^2 are member of the sequence.
a(2) = 31 = p^2+3pq+q^2 for p=2, q=3.
a(25) = 671 = p^2+3pq+q^2 for (p,q)=(2,23) and (5,19), is the least term to allow more than 1 decomposition.
a(1431) = 136895 = p^2+3pq+q^2 for (p,q)=(2,367), (67,277) and (103,233), is the least term to allow more than 2 decompositions.

Mathematica

nf[{a_, b_}]:=a^2+3a*b+b^2; Take[Union[nf/@Tuples[Prime[Range[20]], 2]], 50] (* Harvey P. Dale, Mar 31 2015 *)

Prog

(PARI) is_A218793(n, v=0)={ /* set v=1 to count number of decompositions, and v=2 to print them */ my(r, c=0); forprime( q=1, sqrtint(n\5), issquare(4*n+5*q^2, &r) || next; isprime((r-3*q)/2) || next; v || return(1); v>1 & print1([q, (r-3*q)/2]", "); c++); c}

Crossrefs

Sequence in context: A158844 A077340 A077343 * A116131 A212050 A075035

Adjacent sequences: A218790 A218791 A218792 * A218794 A218795 A218796

Keyword

nonn

Author

M. F. Hasler, Nov 05 2012

Status

approved