A233346 Primes of the form p(k)^2 + q(m)^2 with k > 0 and m > 0, where p(.) is the partition function (A000041), and q(.) is the strict partition function (A000009).

2, 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 101, 109, 113, 137, 149, 157, 193, 229, 241, 349, 373, 509, 709, 733, 1033, 1049, 1213, 1249, 1453, 1493, 1669, 1789, 2141, 2237, 2341, 2917, 3037, 3137, 3361, 4217, 5801, 5897, 6029, 6073, 8821, 10301, 10937, 11057, 18229, 18289, 19249, 20173, 20341, 20389, 21017, 24001, 30977, 36913, 42793

Offset: 1

Author

Zhi-Wei Sun, Dec 07 2013

Keywords

• nonn

Comments

Conjecture: The sequence contains infinitely many terms.

This follows from part (i) of the conjecture in A233307. Similarly, the conjecture in A232504 implies that there are infinitely many primes of the form p(k) + q(m) with k and m positive integers.

Examples

			a(1) = 2 since p(1)^2 + q(1)^2 = 1^2 + 1^2 = 2.
a(2) = 5 since p(1)^2 + q(3)^2 = 1^2 + 2^2 = 5.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..176
• Z.-W. Sun, On a^n+ bn modulo m, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.

Crossrefs

Cf. A000009, A000040, A000041, A002313, A232504, A233307.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
n=0
Do[If[Mod[Prime[m]+1,4]>0,Do[If[PartitionsP[j]>=Sqrt[Prime[m]],Goto[aa],
If[SQ[Prime[m]-PartitionsP[j]^2]==False,Goto[bb],Do[If[PartitionsQ[k]^2==Prime[m]-PartitionsP[j]^2,
n=n+1;Print[n," ",Prime[m]];Goto[aa]];If[PartitionsQ[k]^2>Prime[m]-PartitionsP[j]^2,Goto[bb]];Continue,{k,1,2*Sqrt[Prime[m]]}]]];
Label[bb];Continue,{j,1,Sqrt[Prime[m]]}]];
Label[aa];Continue,{m,1,4475}]