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

1,1

Comments

Conjecture: The sequence contains infinitely many terms.

This follows from part (i) of the conjecture in A223307. 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]]}]]];
Label[bb]; Continue, {j, 1, Sqrt[Prime[m]]}]];
Label[aa]; Continue, {m, 1, 4475}]

Crossrefs

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

Sequence in context: A135933 A086807 A002313 * A182198 A291275 A291278

Adjacent sequences: A233343 A233344 A233345 * A233347 A233348 A233349

Keyword

nonn

Author

Zhi-Wei Sun, Dec 07 2013

Status

approved