A218771 -id:A218771 - cp's OEIS frontend

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

M. F. Hasler, Nov 05 2012

nonn

Sequence A218771 is the subsequence of primes in this sequence.

			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.

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

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

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}

A218794 Numbers that can be written as p^2 + 3pq + q^2 with primes p < q.

Original entry on oeis.org

31, 59, 79, 95, 121, 179, 191, 229, 251, 295, 311, 389, 395, 401, 451, 479, 491, 541, 569, 671, 695, 719, 745, 809, 899, 971, 1019, 1061, 1109, 1111, 1121, 1151, 1249, 1271, 1301, 1409, 1451, 1499, 1595, 1619, 1661, 1711, 1919, 1931, 1949, 1991, 2059, 2105, 2111, 2141, 2195, 2201, 2245

Offset: 1

M. F. Hasler, Nov 05 2012

nonn

This is a subsequence of A218793, with the restriction that p < q, excluding terms of the form 5p^2 unless they would have another decomposition of the given form.

Sequence A218771 is the subsequence of primes in this sequence.

			a(1) = 31 = p^2+3pq+q^2 for p=2, q=3.
a(20) = 671 = p^2+3pq+q^2 for (p,q)=(2,23) and (5,19) is the least term to allow more than 1 decomposition. See A218795 for more such terms.

With[{nn=60},Take[Union[#[[1]]^2+3Times@@#+#[[2]]^2&/@Subsets[Prime[ Range[ Floor[nn/3]]],{2}]],nn]] (* Harvey P. Dale, Apr 08 2013 *)

is_A218794(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-1)\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}

A218795 Numbers that can be written in more than 1 way as p^2 + 3pq + q^2 with primes p < q.

Original entry on oeis.org

671, 1595, 3629, 3839, 4661, 4895, 5771, 8459, 11495, 13079, 17951, 19829, 21251, 22895, 23411, 34751, 40211, 40799, 43829, 45791, 46031, 47795, 49001, 53009, 57209, 57959, 60119, 60491, 64829, 67595, 70661, 78755, 83201, 89249, 98879

Offset: 1

M. F. Hasler, Nov 05 2012

nonn

According to the comment in A218771, there is no prime in this sequence.

The subsequence of terms allowing more than 2 decompositions of the given form starts 136895, 266741, 716441, 902495, 941501, 963281, 1003295, 1033241, 1236539, 1245431, 1439345, 1463495, 1692995, 2300045, 2430461, 3016145, 3238289,...

			a(1) = 671 = p^2+3pq+q^2 for (p,q)=(2,23) and (5,19), is the least term to allow more than 1 decomposition of the given form.

Cf. A218794 for PARI code and further discussion.

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

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A218794 Numbers that can be written as p^2 + 3pq + q^2 with primes p < q.

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A218795 Numbers that can be written in more than 1 way as p^2 + 3pq + q^2 with primes p < q.

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