A233346 - OEIS

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

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OFFSET

1,1

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.

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.

EXAMPLE

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.

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]]}]]];