A238516 -id:A238516 - cp's OEIS frontend

A238393 a(n) = |{0 < k <= n: 2p(k)p(n) + 1 is prime}|, where p(.) is the partition function (A000041).

1, 1, 3, 2, 3, 2, 5, 2, 6, 3, 3, 5, 1, 9, 3, 4, 5, 5, 6, 2, 7, 3, 5, 8, 3, 4, 8, 10, 7, 10, 6, 7, 9, 8, 8, 6, 6, 4, 12, 10, 10, 8, 6, 6, 5, 7, 8, 6, 10, 5, 9, 9, 11, 7, 7, 6, 9, 11, 8, 7, 11, 6, 9, 8, 4, 8, 5, 18, 14, 10, 9, 7, 8, 6, 13, 9, 4, 7, 7, 15

Offset: 1

Zhi-Wei Sun, Mar 01 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 2, 13.
(ii) If n > 1, then 2*p(k)*p(n) - 1 is prime for some 0 < k < n.
(iii) For any integer n > 0, p(k)*(p(n)+1) + 1 is prime for some k = 1, ..., n.

			a(2) = 1 since 2*p(1)*p(2) + 1 = 2*1*2 + 1 = 5 is prime.
a(13) = 1 since 2*p(3)*p(13) + 1 = 2*3*101 + 1 = 607 is prime.

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

Cf. A000040, A000041, A238457, A238509, A238516.

p[n_,k_]:=PrimeQ[2*PartitionsP[n]*PartitionsP[k]+1]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,n}]
Table[a[n],{n,1,80}]

A239209 a(n) = |{0 < k < n: kp(n)(p(n)-1) + 1 is prime}|, where p(.) is the partition function (A000041).

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Mar 12 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 1. If n > 1 is not equal to 25, then k*p(n)*(p(n)+1) - 1 is prime for some 0 < k < n.
(ii) Let q(.) be the strict partition function given by A000009. Then, for any integer n > 2, there is a number k among 1, ..., n with k*q(n)^2 - 1 prime. Also, we may replace k*q(n)^2 - 1 by k*q(n)^2 + 1 or k*q(n)*(q(n)+1) + 1 or k*q(n)*(q(n)+1) - 1.
We have verified that a(n) > 0 for all n = 2..10^5.

			a(2) = 1 since 1*p(2)*(p(2)-1) + 1 = 1*2*1 + 1 = 3 is prime.
a(6) = 1 since 3*p(6)*(p(6)-1) + 1 = 3*11*10 + 1 = 331 is prime.
a(69) = 1 since 50*p(69)*(p(69)-1) + 1 =  50*3554345*3554344 + 1 = 631668241234001 is prime.
a(76) = 1 since 24*p(76)*(p(76)-1) + 1 = 24*9289091*9289090 + 1 = 2070892855612561 is prime.

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

Cf. A000009, A000040, A000041, A238393, A238457, A238458, A238459, A238509, A238516, A238577, A239207, A239214.

p[n_]:=PartitionsP[n]
f[n_]:=p[n]*(p[n]-1)
a[n_]:=Sum[If[PrimeQ[k*f[n]+1],1,0],{k,1,n-1}]
Table[a[n],{n,1,80}]

A239214 a(n) = |{0 < k < n: p(k)p(n)(p(n)+1) - 1 is prime}|, where p(.) is the partition function (A000041).

Original entry on oeis.org

0, 1, 2, 3, 1, 3, 3, 2, 3, 3, 5, 4, 4, 3, 3, 6, 2, 4, 5, 4, 1, 2, 3, 6, 6, 6, 2, 4, 6, 9, 2, 7, 8, 6, 6, 2, 2, 2, 10, 4, 4, 7, 5, 7, 1, 4, 9, 9, 9, 4, 6, 8, 7, 8, 6, 4, 13, 10, 3, 6, 10, 7, 13, 12, 12, 8, 6, 8, 5, 11, 5, 3, 4, 5, 11, 7, 6, 12, 16, 4

Offset: 1

Zhi-Wei Sun, Mar 12 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 1.
(ii) For each n = 2, 3, ... there is a positive integer k < n with p(k)*p(n)*(p(n)-1) + 1 prime. If n > 2, then p(k)*p(n)*(p(n)-1)-1 is prime for some 0 < k < n.
(iii) For any n > 1, there is a positive integer k < n with 2*p(k)*p(n)*A000009(n)*A047967(n) + 1 prime.
We have verified that a(n) > 0 for all n = 2, ..., 10^5.

			a(2) = 1 since p(1)*p(2)*(p(2)+1) - 1 = 1*2*3 - 1 = 5 is prime.
a(5) = 1 since p(3)*p(5)*(p(5)+1) - 1 = 3*7*8 - 1 = 167 is prime.
a(21) = 1 since p(10)*p(21)*(p(21)+1) - 1 = 42*792*793 - 1 = 26378351 is prime.
a(45) = 1 since p(20)*p(45)*(p(45)+1) - 1 = 627*89134*89135 - 1 = 4981489349429 is prime.

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

Cf. A000040, A000041, A238457, A238509, A238516, A238393, A239207, A239209.

p[n_]:=PartitionsP[n]
f[n_]:=p[n]*(p[n]+1)
a[n_]:=Sum[If[PrimeQ[p[k]*f[n]-1],1,0],{k,1,n-1}]
Table[a[n],{n,1,80}]

A238577 a(n) = |{0 < k <= n: p(n)q(k)r(k) + 1 is prime}|, where p(.), q(.) and r(.) are given by A000041, A000009 and A047967 respectively.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Mar 01 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 3, 5, 20. If n > 2, then p(n)*q(k)*r(k) - 1 is prime for some k = 1, ..., n.

(ii) If n > 2 is not equal to 22, then p(n)*q(n)*q(k) - 1 is prime for some k = 1, ..., n. If n > 13, then p(n)*q(k)*q(n-k) - 1 is prime for some 1 < k < n/2.

			a(5) = 1 since p(5)*q(4)*r(4) + 1 = 7*2*3 + 1 = 43 is prime.
a(20) = 1 since p(20)*q(13)*r(13) + 1 = 627*18*83 + 1 = 936739 is prime.

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

Cf. A000009, A000040, A000041, A047967, A236442, A238457, A238509, A238393, A238516.

p[n_,k_]:=PrimeQ[PartitionsP[n]*PartitionsQ[k]*(PartitionsP[k]-PartitionsQ[k])+1]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,n}]
Table[a[n],{n,1,80}]

A239232 a(n) = |{0 < k <= n: p(n+k) + 1 is prime}|, where p(.) is the partition function (A000041).

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Mar 13 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 3.
(ii) If n > 15, then p(n+k) - 1 is prime for some k = 1, ..., n.
(iii) If n > 38, then p(n+k) is prime for some k = 1, ..., n.
The conjecture implies that there are infinitely many positive integers m with p(m) + 1 (or p(m) - 1, or p(m)) prime.

			a(4) = 1 since p(4+4) + 1 = 22 + 1 = 23 is prime.
a(8) = 2 since p(8+1) + 1 = 31 and p(8+2) + 1 = 43 are both prime.
a(11) = 1 since p(11+8) + 1 = 491 is prime.

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

Cf. A000040, A000041, A049575, A234470, A234569, A238393, A238457, A238509, A238516, A239207, A239209, A239214.

p[n_]:=PartitionsP[n]
a[n_]:=Sum[If[PrimeQ[p[n+k]+1],1,0],{k,1,n}]
Table[a[n],{n,1,80}]

cp's OEIS Frontend

A238393 a(n) = |{0 < k <= n: 2*p(k)*p(n) + 1 is prime}|, where p(.) is the partition function (A000041).

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A239209 a(n) = |{0 < k < n: k*p(n)*(p(n)-1) + 1 is prime}|, where p(.) is the partition function (A000041).

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A239214 a(n) = |{0 < k < n: p(k)*p(n)*(p(n)+1) - 1 is prime}|, where p(.) is the partition function (A000041).

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A238577 a(n) = |{0 < k <= n: p(n)*q(k)*r(k) + 1 is prime}|, where p(.), q(.) and r(.) are given by A000041, A000009 and A047967 respectively.

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A239232 a(n) = |{0 < k <= n: p(n+k) + 1 is prime}|, where p(.) is the partition function (A000041).

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Mathematica

A238393 a(n) = |{0 < k <= n: 2p(k)p(n) + 1 is prime}|, where p(.) is the partition function (A000041).

A239209 a(n) = |{0 < k < n: kp(n)(p(n)-1) + 1 is prime}|, where p(.) is the partition function (A000041).

A239214 a(n) = |{0 < k < n: p(k)p(n)(p(n)+1) - 1 is prime}|, where p(.) is the partition function (A000041).

A238577 a(n) = |{0 < k <= n: p(n)q(k)r(k) + 1 is prime}|, where p(.), q(.) and r(.) are given by A000041, A000009 and A047967 respectively.