A236417 -id:A236417 - 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}]

A236419 a(n) = |{0 < k < n: r = phi(k) + phi(n-k)/6 + 1 and p(r) + q(r) are both prime}|, where phi(.) is Euler's totient function, p(.) is the partition function (A000041) and q(.) is the strict partition function (A000009).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 2, 1, 0, 1, 0, 2, 0, 1, 1, 0, 0, 4, 1, 2, 1, 0, 2, 3

Offset: 1

Zhi-Wei Sun, Jan 25 2014

nonn

Conjecture: a(n) > 0 for all n > 127.
We have verified this for n up to 30000.
The conjecture implies that there are infinitely many primes r with p(r) + q(r) also prime.

			a(15) = 1 since phi(1) + phi(14)/6 + 1 = 3 with p(3) + q(3) = 3 + 2 = 5 prime.
a(54) = 1 since phi(41) + phi(13)/6 + 1 = 43 with p(43) + q(43) = 63261 + 1610 = 64871 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 [math.NT], 2014.

Cf. A000009, A000010, A000040, A232504, A236412, A236417.

pq[n_]:=PrimeQ[n]&&PrimeQ[PartitionsP[n]+PartitionsQ[n]]
f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/6+1
a[n_]:=Sum[If[pq[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

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 *)

A236439 a(n) = |{0 < k < n-2: A000009(m)^2 + A047967(m)^2 is prime with m = k + phi(n-k)/2}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 3, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 4, 2, 3, 2, 3, 5, 4, 3, 2, 6, 6, 4, 2, 1, 8, 4, 4, 3, 1, 6, 4, 3, 3, 3, 3, 3, 4, 4, 5, 3, 4, 5, 3, 3, 7, 4, 5, 5, 5, 11, 7, 6, 3, 7, 8, 6, 5, 5, 8, 6, 7, 11, 7, 5, 7, 8, 7, 7, 5, 10, 10, 5, 6, 8, 6, 10, 8, 6, 8, 11, 10, 6, 10, 7, 7, 9, 4, 9, 11, 8, 13, 7

Offset: 1

Zhi-Wei Sun, Jan 25 2014

nonn

Conjecture: a(n) > 0 for all n > 3.
We have verified this for n up to 50000.
The conjecture implies that there are infinitely many positive integers m with A000009(m)^2 + A047967(m)^2 prime. See A236440 for such numbers m.

			a(14) = 1 since 2 + phi(12)/2 = 4 with A000009(4)^2 + A047967(4)^2 = 2^2 + 3^2 = 13 prime.
a(17) = 1 since 10 + phi(7)/2 = 13 with A000009(13)^2 + A047967(13)^2 = 18^2 + 83^2 = 7213 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. A000009, A000010, A000040, A047967, A236412, A236417, A236419, A236440.

p[n_]:=PrimeQ[PartitionsQ[n]^2+(PartitionsP[n]-PartitionsQ[n])^2]
a[n_]:=Sum[If[p[k+EulerPhi[n-k]/2],1,0],{k,1,n-3}]
Table[a[n],{n,1,100}]

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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A236419 a(n) = |{0 < k < n: r = phi(k) + phi(n-k)/6 + 1 and p(r) + q(r) are both prime}|, where phi(.) is Euler's totient function, 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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A236439 a(n) = |{0 < k < n-2: A000009(m)^2 + A047967(m)^2 is prime with m = k + phi(n-k)/2}|, where phi(.) is Euler's totient function.

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