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
Keywords
Examples
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.
Links
- M. F. Hasler, Table of n, a(n) for n = 1..7196 (all terms below 10^6).
Programs
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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 *) -
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}
Comments