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 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). 17
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified August 19 20:20 EDT 2019. Contains 326133 sequences. (Running on oeis4.)