A236413 -id:A236413 - cp's OEIS frontend

A236412 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/8 is an integer with p(m)^2 + q(m)^2 prime}|, where phi(.) is Euler's totient, p(.) is the partition function (A000041) and q(.) is the strict partition function (A000009).

0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 0, 0, 1, 2, 3, 3, 4, 5, 3, 4, 4, 7, 4, 5, 5, 3, 3, 4, 5, 4, 3, 6, 8, 3, 3, 3, 7, 3, 7, 4, 5, 3, 6, 3, 2, 3, 6, 3, 3, 2, 5, 1, 4, 6, 4, 3, 3, 7, 5, 3, 3, 3, 4, 1, 5, 4, 3, 2, 4, 3, 6, 2, 5, 6, 4, 5, 2, 1, 6, 4, 4, 2, 11, 1, 6, 3, 5, 6, 7, 2, 4, 4, 2, 3, 2

Offset: 1

Zhi-Wei Sun, Jan 24 2014

nonn

Conjecture: a(n) > 0 for all n > 17.
We have verified this for n up to 65000.
The conjecture implies that there are infinitely positive integers m with p(m)^2 + q(m)^2 prime. See A236413 for a list of such numbers m. See also A236414 for primes of the form p(m)^2 + q(m)^2.

			a(15) = 1 since phi(2)/2 + phi(13)/8 = 1/2 + 12/8 = 2 with p(2)^2 + q(2)^2 = 2^2 + 1^2 = 5 prime.
a(69) = 1 since phi(5)/2 + phi(64)/8 = 2 + 4 = 6 with p(6)^2 + q(6)^2 = 11^2 + 4^2 = 137 prime.
a(89) = 1 since phi(73)/2 + phi(16)/8 = 36 + 1 = 37 with p(37)^2 + q(37)^2 = 21637^2 + 760^2 = 468737369 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000010, A000040, A000041, A233307, A236374, A236413, A236414.

p[n_]:=IntegerQ[n]&&PrimeQ[PartitionsP[n]^2+PartitionsQ[n]^2]
f[n_,k_]:=EulerPhi[k]/2+EulerPhi[n-k]/8
a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A236440 Positive integers m with A000009(m)^2 + A047967(m)^2 prime.

Original entry on oeis.org

2, 3, 4, 13, 18, 23, 44, 52, 54, 67, 82, 93, 139, 155, 166, 185, 196, 249, 299, 333, 382, 559, 574, 911, 939, 1076, 1077, 1386, 1707, 1710, 1872, 2041, 2120, 2172, 2234, 2810, 3272, 3407, 3442, 3469, 3551, 3657, 3694, 4185, 4282, 4469, 4554, 5273, 5315, 5729

Offset: 1

Zhi-Wei Sun, Jan 25 2014

nonn

According to the conjecture in A236439, this sequence should have infinitely many terms.

			a(1) = 2 since A000009(2)^2 + A047967(2)^2 = 1^2 + 1^2 =2 is prime, but A000009(1)^2 + A047967(1)^2 = 1^2 + 0^2 is not.

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, A000040, A047967, A236413, A236418, A236439.

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

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}]

A236418 Primes p with A047967(p) also prime.

Original entry on oeis.org

13, 23, 43, 53, 71, 83, 107, 257, 269, 313, 1093, 2659, 2851, 3527, 8243, 20173, 20717, 24329, 26161, 26237, 31583, 53611, 60719, 74717, 83401, 118259, 118369, 130817, 133811, 145109, 152381, 169111, 178613, 183397, 205963

Offset: 1

Zhi-Wei Sun, Jan 25 2014

nonn

According to the conjecture in A236417, this sequence should have infinitely many terms.

			a(1) = 13 with 13 and A047967(13) = 83 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..42

Cf. A000040, A047967, A234530, A234569, A234644, A235344, A235346, A236413, A236417, A236419, A236440.

pq[n_]:=PrimeQ[n]&&PrimeQ[PartitionsP[n]-PartitionsQ[n]]
n=0;Do[If[pq[m],n=n+1;Print[n," ",m]],{m,1,10000}]
Select[Prime[Range[20000]],PrimeQ[PartitionsP[#]-PartitionsQ[#]]&] (* Harvey P. Dale, Jan 02 2022 *)

cp's OEIS Frontend

A236412 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/8 is an integer with p(m)^2 + q(m)^2 prime}|, where phi(.) is Euler's totient, p(.) is the partition function (A000041) and q(.) is the strict partition function (A000009).

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A236440 Positive integers m with A000009(m)^2 + A047967(m)^2 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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A236418 Primes p with A047967(p) also prime.

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