A238458 -id:A238458 - cp's OEIS frontend

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

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}]

A238459 Number of primes p < n with q(n-p) + 1 prime, where q(.) is the strict partition function (A000009).

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Feb 27 2014

nonn

Conjecture: a(n) > 0 for all n > 2. Also, for each n > 6 there is a prime p < n with q(n-p) - 1 prime.
We have verified the conjecture for n up to 10^5.
See also A238458 for a similar conjecture involving the partition function p(n).

			a(3) = 1 since 2 and q(3-2) + 1 = 1 + 1 = 2 are both prime.
a(28) = 1 since 17 and q(28-17) + 1 = q(11) + 1 = 12 + 1 = 13 are both 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, A237705, A237768, A237769, A238457, A238458.

q[n_,k_]:=PrimeQ[PartitionsQ[n-Prime[k]]+1]
a[n_]:=Sum[If[q[n,k],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,100}]

cp's OEIS Frontend

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

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A238459 Number of primes p < n with q(n-p) + 1 prime, where q(.) is the strict partition function (A000009).

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cp's OEIS Frontend

A239209 a(n) = |{0 < k < n: 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

A238459 Number of primes p < n with q(n-p) + 1 prime, where q(.) is the strict partition function (A000009).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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