A239209 -id:A239209 - cp's OEIS frontend

A239207 a(n) = |{0 < k <= n: kp(n)q(n)*r(n) - 1 is prime}|, where p(.), q(.) and r(.) are given by A000041, A000009 and A047967 respectively.

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

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)*q(n)*r(n) + 1 is prime for some k = 1, ..., n.
(ii) For any integer n > 1, there is a number k among 1, ..., n with k*p(n)*q(n) - 1 (or k*p(n)*q(n) + 1) prime.
(iii) For each n > 1, there is a positive integer k < n with k*p(n) + 1 (or k*q(n) + 1) prime. If n > 1, then k*p(n) - 1 is prime for some k = 1, ..., n. If n > 2, then k*q(n) - 1 is prime for some 0 < k < n.
We have verified that a(n) > 0 for all n = 2, ..., 83000.
See also A239209 and A239214 for related conjectures.

			a(2) = 1 since 2*p(2)*q(2)*r(2) - 1 = 2*2*1*1 - 1 = 3 is prime.
a(23) = 1 since 12*p(23)*q(23)*r(23) - 1 = 12*1255*104*1151 - 1 = 1802742239 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, A047967, A238393, A238577, A239209, A239214.

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

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

A239207 a(n) = |{0 < k <= n: kp(n)q(n)*r(n) - 1 is prime}|, where p(.), q(.) and r(.) are given by A000041, A000009 and A047967 respectively.

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

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cp's OEIS Frontend

A239207 a(n) = |{0 < k <= n: k*p(n)*q(n)*r(n) - 1 is prime}|, where p(.), q(.) and r(.) are given by A000041, A000009 and A047967 respectively.

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Comments

Examples

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Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A239207 a(n) = |{0 < k <= n: kp(n)q(n)*r(n) - 1 is prime}|, where p(.), q(.) and r(.) are given by A000041, A000009 and A047967 respectively.

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