A233346 -id:A233346 - cp's OEIS frontend

A236413 Positive integers m with p(m)^2 + q(m)^2 prime, where p(.) is the partition function (A000041) and q(.) is the strict partition function (A000009).

1, 2, 3, 4, 6, 17, 24, 37, 44, 95, 121, 162, 165, 247, 263, 601, 714, 742, 762, 804, 1062, 1144, 1149, 1323, 1508, 1755, 1833, 1877, 2330, 2380, 2599, 3313, 3334, 3368, 3376, 3395, 3504, 3688, 3881, 4294, 4598, 4611, 5604, 5696, 5764, 5988, 6552, 7206, 7540, 7689

Offset: 1

Zhi-Wei Sun, Jan 24 2014

nonn

According to the conjecture in A236412, this sequence should have infinitely many terms.
See A236414 for primes of the form p(m)^2 + q(m)^2.
See also A236440 for a similar sequence.

			a(1) = 1 since p(1)^2 + q(1)^2 = 1^2 + 1^2 = 2 is prime.
a(2) = 2 since p(2)^2 + q(2)^2 = 2^2 + 1^2 = 5 is prime.
a(3) = 3 since p(3)^2 + q(3)^2 = 3^2 + 2^2 = 13 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..200
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000009, A000010, A000040, A233346, A236412, A236414, A236417, A236418, A236419, A236440.

pq[n_]:=PrimeQ[PartitionsP[n]^2+PartitionsQ[n]^2]
n=0;Do[If[pq[m],n=n+1;Print[n," ",m]],{m,1,10000}]

A233360 Primes of the form L(k) + q(m) with k > 0 and m > 0, where L(k) is the k-th Lucas number (A000204), and q(.) is the strict partition function (A000009).

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 79, 83, 101, 103, 107, 127, 131, 149, 151, 193, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 307, 337, 347, 349, 379, 397, 401, 419, 421, 449, 463, 487, 523, 541, 571, 643, 647, 661

Offset: 1

Zhi-Wei Sun, Dec 08 2013

nonn

Conjecture: The sequence has infinitely many terms.

This follows from the conjecture in A233359.

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

Zhi-Wei Sun, Table of n, a(n) for n = 1..270

Cf. A000009, A000040, A000032, A000204, A232504, A233307, A233346, A233359.

n=0
Do[Do[If[LucasL[j]>=Prime[m],Goto[aa],
Do[If[PartitionsQ[k]==Prime[m]-LucasL[j],
n=n+1;Print[n," ",Prime[m]];Goto[aa]];If[PartitionsQ[k]>Prime[m]-LucasL[j],Goto[bb]];Continue,{k,1,2*(Prime[m]-LucasL[j])}]];
Label[bb];Continue,{j,1,2*Log[2,Prime[m]]}];
Label[aa];Continue,{m,1,125}]

A233439 a(n) = |{0 < k < n: prime(k)^2 + 4*prime(n-k)^2 is prime}|.

Original entry on oeis.org

0, 0, 0, 1, 2, 1, 2, 1, 3, 4, 4, 8, 4, 6, 3, 1, 7, 3, 8, 5, 2, 9, 2, 11, 8, 7, 5, 4, 8, 7, 8, 8, 8, 7, 5, 9, 5, 10, 9, 7, 13, 9, 11, 10, 14, 5, 11, 10, 10, 11, 12, 7, 13, 10, 10, 8, 15, 11, 12, 11, 13, 14, 6, 12, 11, 22, 21, 5, 15, 7, 13, 15, 17, 15, 10, 16, 11, 13, 14, 12, 17, 12, 16, 16, 19, 22, 17, 12, 19, 17, 19, 17, 16, 17, 18, 20, 19, 17, 10, 16

Offset: 1

Zhi-Wei Sun, Dec 09 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 3.
(ii) For any integer n > 10, prime(j)^3 + 2*prime(n-j)^2 is prime for some 0 < j < n, and prime(k)^3 + 2*prime(n-k)^3 is prime for some 0 < k < n.
(iii) If n > 5, then prime(k)^3 + 2*p(n-k)^3 is prime for some 0 < k < n, where p(.) is the partition function (A000041). If n > 2, then prime(k)^3 + 2*q(n-k)^3 is prime for some 0 < k < n, where q(.) is the strict partition function (A000009).

			a(4) = 1 since prime(3)^2 + 4*prime(1)^2 = 5^2 + 4*2^2 = 41 is prime.
a(6) = 1 since prime(5)^2 + 4*prime(1)^2 = 11^2 + 4*2^2 = 137 is prime.
a(8) = 1 since prime(3)^2 + 4*prime(5)^2 = 5^2 + 4*11^2 = 509 is prime.
a(16) = 1 since prime(6)^2 + 4*prime(10)^2 = 13^2 + 4*29^2 = 3533 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Z.-W. Sun, On a^n+ bn modulo m, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2016.

Cf. A000040, A002313, A220413, A232174, A232269, A232465, A232502, A233150, A233204, A233206, A233296, A233307, A233346.

a[n_]:=Sum[If[PrimeQ[Prime[k]^2+4*Prime[n-k]^2],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A236414 Primes of the form p(m)^2 + q(m)^2 with m > 0, where p(.) is the partition function (A000041) and q(.) is the strict partition function (A000009).

Original entry on oeis.org

2, 5, 13, 29, 137, 89653, 2495509, 468737369, 5654578481, 10952004689145437, 4227750418844538601, 16877624537532512753869, 29718246090638680022401, 33479444420637044862046313837, 386681772864767371008755193761

Offset: 1

Zhi-Wei Sun, Jan 24 2014

nonn

This is a subsequence of A233346. All terms after the first term are congruent to 1 modulo 4.

According to the conjecture in A236412, this sequence should have infinitely many terms. See A236413 for positive integers m with p(m)^2 + q(m)^2 prime.

			a(1) = 2 since 2 = p(1)^2 + q(1)^2 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..50

Cf. A000009, A000010, A000040, A000041, A233346, A236412, A236413.

a[n_]:=PartitionsP[A236413(n)]^2+PartitionsQ[A236413(n)]^2
Table[a[n],{n,1,15}]

A229835 Number of ways to write n = (p - 1)/6 + q, where p is a prime, and q is a term of the sequence A000009.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 5, 5, 5, 4, 6, 5, 7, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 10, 9, 6, 8, 6, 10, 8, 9, 7, 7, 10, 10, 9, 8, 7, 10, 7, 10, 3, 7, 12, 8, 10, 6, 8, 9, 6, 10, 8, 11, 7, 11, 8, 7, 9, 8, 12, 10, 8, 12, 7, 9, 10, 10, 8, 11, 10, 7, 10, 9, 14, 9, 9, 9, 8, 10, 10, 9, 7, 8, 9, 9, 8, 10, 9, 10, 10, 9, 7, 8, 7, 12, 8

Offset: 1

Zhi-Wei Sun, Dec 19 2013

nonn

Conjecture: a(n) > 0 for all n > 1. Also, any integer n > 1 can be written as (p + 1)/6 + q, where p is a prime and q is a term of A000009.

We have verified this for n up to 2*10^8. Note that 26128189 cannot be written as (p - 1)/4 + q with p a prime and q a term of A000009. Also, 65152682 cannot be written as (p + 1)/4 + q with p a prime and q a term of A000009.

			a(2) = 1 since 2 = (7 - 1)/ 6 + 1 with 7 prime, and 1 = A000009(i) for i = 0, 1, 2.
a(3) = 2 since 3 = (7 - 1 )/6 + 2 with 7 prime and 2 = A000009(3) = A000009(4), and 3 = (13 - 1 )/6 + 1 with 13 prime and 1 = A000009(i) for i = 0, 1, 2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000009, A000040, A232398, A232463, A233307, A233346, A233390, A233393, A233417, A233359.

Do[m=0;Do[If[PartitionsQ[k]>=n,Goto[aa]];If[k>1&&PartitionsQ[k]==PartitionsQ[k-1],Goto[bb]];
If[PrimeQ[6(n-PartitionsQ[k])+1],m=m+1];Label[bb];Continue,{k,1,2n}];
Label[aa];Print[n," ",m];Continue,{n,1,100}]

A234900 Primes p with P(p+1) also prime, where P(.) is the partition function (A000041).

Original entry on oeis.org

2, 3, 5, 131, 167, 211, 439, 2731, 3167, 3541, 4261, 7457, 8447, 18289, 22669, 23201, 23557, 35401, 44507, 76781, 88721, 108131, 126097, 127079, 136319, 141359, 144139, 159169, 164089, 177487, 202627, 261757, 271181, 282911, 291971, 307067, 320561, 389219, 481589, 482627, 602867, 624259, 662107, 682361, 818887, 907657, 914189, 964267, 1040191, 1061689

Offset: 1

Zhi-Wei Sun, Jan 01 2014

nonn

It seems that this sequence contains infinitely many terms.

See also A234569 for a similar sequence.

			a(1) = 2 since P(2+1) = 3 is prime.
a(2) = 3 since P(3+1) = 5 is prime.
a(3) = 5 since P(5+1) = 11 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..60

Cf. A000040, A000041, A049575, A233346, A234470, A234514, A234530, A234567, A234569, A234615, A234644.

p[k_]:=p[k]=PrimeQ[PartitionsP[Prime[k]+1]]
n=0;Do[If[p[k],n=n+1;Print[n," ",Prime[k]]],{k,1,10000}]

cp's OEIS Frontend

A236413 Positive integers m with p(m)^2 + q(m)^2 prime, where p(.) is the partition function (A000041) and q(.) is the strict partition function (A000009).

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A233360 Primes of the form L(k) + q(m) with k > 0 and m > 0, where L(k) is the k-th Lucas number (A000204), and q(.) is the strict partition function (A000009).

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A233439 a(n) = |{0 < k < n: prime(k)^2 + 4*prime(n-k)^2 is prime}|.

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A236414 Primes of the form p(m)^2 + q(m)^2 with m > 0, where p(.) is the partition function (A000041) and q(.) is the strict partition function (A000009).

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A229835 Number of ways to write n = (p - 1)/6 + q, where p is a prime, and q is a term of the sequence A000009.

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A234900 Primes p with P(p+1) also prime, where P(.) is the partition function (A000041).

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