A233183 -id:A233183 - cp's OEIS frontend

A234694 a(n) = |{0 < k < n: p = k + prime(n-k) and prime(p) - p + 1 are both prime}|.

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

Offset: 1

Zhi-Wei Sun, Dec 29 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 9. Also, for any integer n > 51 there is a positive integer k < n such that p = k + prime(n-k) and prime(p) + p + 1 are both prime.
(ii) If n > 9 (or n > 21), then there is a positive integer k < n such that m - 1 and prime(m) + m (or prime(m) - m, resp.) are both prime, where m = k + prime(n-k).
(iii) If n > 483, then for some 0 < k < n both prime(m) + m and prime(m) - m are prime, where m = k + prime(n-k).
(iv) If n > 3, then there is a positive integer k < n such that prime(k + prime(n-k)) + 2 is prime.
Clearly, part (i) of the conjecture implies that there are infinitely many primes p with prime(p) - p + 1 (or prime(p) + p + 1) also prime.
See A234695 for primes p with prime(p) - p + 1 also prime.

			a(5) = 1 since 2 + prime(3) = 7 and prime(7) - 6 = 11 are both prime.
a(25) = 1 since 20 + prime(5) = 31 and prime(31) - 30 = 97 are both prime.
a(27) = 1 since 18 + prime(9) = 41 and prime(41) - 40 = 139 are both prime.
a(45) = 1 since 6 + prime(39) = 173 and prime(173) - 172 = 859 are both prime.
a(49) = 1 since 26 + prime(23) = 109 and prime(109) - 108 = 491 are both 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. A000040, A014688, A014689, A014692, A064269, A064270, A232861, A233150, A233183, A233206, A233296, A234695.

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

A233150 Number of ways to write n = k + m (k, m > 0) with 2^k + prime(m) prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Dec 05 2013

nonn

Conjecture: a(n) > 0 except for n = 1, 2, 7.

We have verified this for n up to 3*10^7. For n = 15687374, the least positive integer k with 2^k + prime(n-k) prime is 51299. For n = 28117716, the least positive integer k with 2^k + prime(n-k) prime is 81539.

			a(9) = 1 since 9 = 7 + 2 with 2^7 + prime(2) = 128 + 3 = 131 prime.
a(13) = 1 since 13 = 3 + 10 with 2^3 + prime(10) = 8 + 29 = 37 prime.
a(588) = 1 since 588 = 66 + 522 with 2^{66} + prime(522) = 2^{66} + 3739 = 73786976294838210203 prime.
a(1012) = 1 since 1012 = 317 + 695 with 2^{317} + prime(695) = 2^{317} + 5231 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..6000
Zhi-Wei Sun, On a^n + b*n modulo m, preprint, arXiv:1312.1166 [math.NT], 2013-2014.
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2016.

Cf. A000040, A000079, A231201, A231557, A231577, A233183, A233204, A233206, A233296.

a[n_]:=Sum[If[PrimeQ[2^k+Prime[n-k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A233204 Number of ways to write n = k + m with 0 < k < m such that 2^k * prime(m) + 3 is prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Dec 05 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 3. Also, any integer n > 2 can be written as k + m (0 < k <= m) with 2^k*prime(m) - 3 prime.

(ii) Any integer n > 6 can be written as k + m (0 < k < m) with prime(k) + 6 and prime(m) + 6 both prime. Each integer n > 4 can be written as k + m (0 < k < m) with prime(k) + 2 and prime(m) + 6 both prime. Also, for every integer n > 3 not among 11, 21, 32, 49, 171, there is a positive integer k < n with prime(k) + 2 and prime(n-k) + 2 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..8000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A000079, A231201, A231561, A233150, A233183.

a(6) = 1 since 6 = 2 + 4 with 2^2*prime(4) + 3 = 4*7 + 3 = 31 prime.
a(22) = 1 since 22 = 1 + 21 with 2^1*prime(21) + 3 = 2*73 + 3 = 149 prime.

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

A233206 Number of ways to write n = k + m (0 < k <= m) with k! + prime(m) prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Dec 05 2013

nonn

Conjecture: a(n) > 0 for all n > 3.

We have verified this for n up to 10^7. For n = 1356199, the least positive integer k with k! + prime(n-k) prime is 4496. For n = 7212995, the smallest positive integer k with k! + prime(n-k) prime is 4507.

			a(6) = 1 since 6 = 3 + 3 with 3! + prime(3) = 6 + 5 = 11 prime.
a(11) = 1 since 11 = 4 + 7 with 4! + prime(7) = 24 + 17 = 41 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A000142, A231201, A231516, A231557, A231561, A231631, A233150, A233183, A233204.

a[n_]:=Sum[If[PrimeQ[k!+Prime[n-k]],1,0],{k,1,n/2}]
Table[a[n],{n,1,100}]

A233296 a(n) = |{0 < k < n: k*prime(n-k) + 1 is prime}|.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 4, 2, 2, 5, 4, 1, 6, 1, 2, 5, 4, 4, 4, 3, 3, 2, 4, 6, 5, 4, 5, 6, 7, 7, 6, 5, 8, 6, 4, 5, 7, 8, 4, 6, 7, 6, 10, 7, 4, 8, 11, 9, 11, 6, 5, 5, 8, 8, 10, 5, 7, 10, 10, 11, 7, 6, 6, 12, 6, 10, 11, 6, 11, 7, 11, 8, 12, 7, 7, 9, 13, 9, 10, 14, 7, 13, 8, 10, 11, 9, 14, 10, 14, 17, 14, 13, 8, 12, 12

Offset: 1

Zhi-Wei Sun, Dec 07 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 1. Similarly, for any integer n > 2, k*prime(n-k) - 1 (or k^2*prime(n-k) - 1) is prime for some 0 < k < n.

(ii) Let n > 3 be an integer. Then k + prime(n-k) is prime for some 0 < k < n. Also, if n is not equal to 13, then k^2 + prime(n-k)^2 is prime for some 0 < k < n.

			a(17) = 1 since 17 = 14 + 3 with 14*prime(3) + 1 = 14*5 + 1 = 71 prime.
a(19) = 1 since 19 = 18 + 1 with 18*prime(1) + 1 = 18*2 + 1 = 37 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Z.-W. Sun, On a^n+ bn modulo m, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2016.

Cf. A000040, A232465, A232502, A233150, A233183, A233204, A233206.

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

A239451 a(n) = |{1 < k < sqrt(n)*log(n): prime(n) + C(prime(k)-1, (prime(k)-1)/2) is prime}|, where C(2j,j) = (2j)!/(j!)^2.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Mar 19 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 4.
(ii) For any integer n > 4, there is an odd prime p < prime(n) with n*C(p-1,(p-1)/2) + 1 prime.
(iii) For every n = 2, 3, ..., there is a positive integer k < 5*sqrt(n)/3 with n*C(2*k,k) - 1 prime.
(iv) For any integer n > 1, k!*n - 1 (or k!*n + 1) is prime for some k = 1, ..., n.
We have verified that a(n) > 0 for all n = 5, ..., 10^7.

			a(6) = 1 since prime(6) + C(prime(3)-1, (prime(3)-1)/2) = 13 + C(4, 2) = 13 + 6 = 19 is prime.
a(8) = 1 since prime(8) + C(prime(5)-1, (prime(5)-1)/2) = 19 + C(10, 5) = 19 + 252 = 271 is prime.
a(14) = 1 since prime(14) + C(prime(6)-1, (prime(6)-1)/2) = 43 + C(12, 6) = 43 + 924 = 967 is prime.
a(7597) = 1 since prime(7597) + C(prime(686)-1, (prime(686)-1)/2) = 77323 + C(5146, 2573) is prime.
a(193407) = 2 since prime(193407) + C(prime(3212)-1, (prime(3212)-1)/2) =  2652113 + C(29586, 14793) and prime(193407) + C(prime(5348)-1, (prime(5348)-1)/2) = 2652113 + C(52312, 26156) are both prime.
a(4517422) > 0 since prime(4517422) + C(prime(6918)-1, (prime(6918)-1)/2) = 77233291 + C(69778, 34889) is prime.
a(4876885) > 0 since prime(4876885) + C(prime(8904)-1, (prime(8904)-1)/2) = 83778493 + C(92202, 46101) is prime.
a(5887242) > 0 since prime(5887242) + C(prime(5678)-1, (prime(5678)-1)/2) = 102316597 + C(55930, 27965) is prime.
a(8000871) > 0 since prime(8000871) + C(prime(4797)-1, (prime(4797)-1)/2) = 141667111 + C(46410, 23205) 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, A000142, A000984, A231516, A233183, A233206.

p[n_,k_]:=PrimeQ[Prime[n]+Binomial[Prime[k]-1,(Prime[k]-1)/2]]
a[n_]:=Sum[If[p[n,k],1,0],{k,2,Ceiling[Sqrt[n]*Log[n]]-1}]
Table[a[n],{n,1,80}]

cp's OEIS Frontend

A234694 a(n) = |{0 < k < n: p = k + prime(n-k) and prime(p) - p + 1 are both prime}|.

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A233150 Number of ways to write n = k + m (k, m > 0) with 2^k + prime(m) prime.

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A233204 Number of ways to write n = k + m with 0 < k < m such that 2^k * prime(m) + 3 is prime.

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A233206 Number of ways to write n = k + m (0 < k <= m) with k! + prime(m) prime.

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A233296 a(n) = |{0 < k < n: k*prime(n-k) + 1 is prime}|.

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A239451 a(n) = |{1 < k < sqrt(n)*log(n): prime(n) + C(prime(k)-1, (prime(k)-1)/2) is prime}|, where C(2j,j) = (2j)!/(j!)^2.

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