A231201 -id:A231201 - cp's OEIS frontend

A232109 Least prime p < n + 5 with n + (p-1)*(p-3)/8 prime, or 0 if such a prime p does not exist.

5, 3, 3, 5, 3, 5, 3, 7, 11, 5, 3, 5, 3, 7, 17, 5, 3, 5, 3, 7, 11, 5, 3, 23, 17, 7, 11, 5, 3, 5, 3, 13, 11, 7, 19, 5, 3, 7, 17, 5, 3, 5, 3, 7, 17, 5, 3, 23, 11, 7, 11, 5, 3, 23, 17, 7, 11, 5, 3, 5, 3, 31, 11, 7, 19, 5, 3, 7, 11, 5, 3, 5, 3, 13, 17, 7, 19, 5, 3, 7, 17, 5, 3, 23, 17, 7, 11, 5, 3, 29, 11, 13, 11, 7, 19, 5, 3, 7, 11, 5

Offset: 1

Zhi-Wei Sun, Nov 18 2013

nonn

Conjecture: a(n) > 0 for all n > 0. Moreover, for any integer n > 1 there exists a prime p < 2*sqrt(n)*log(7n) such that n + (p-1)*(p-3)/8 is prime.

This implies that any integer n > 1 can be written as (p-1)/2 + q with q a positive integer, and p and (p^2-1)/8 + q both prime.

			a(1) = 5 since neither 1 + (2-1)*(2-3)/8 = 7/8 nor 1 + (3-1)*(3-3)/8 = 1  is prime, but 1 + (5-1)*(5-3)/8 = 2 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A185636, A204065, A227898, A228425, A231201, A231557.

Do[Do[If[PrimeQ[n+(Prime[k]-1)(Prime[k]-3)/8],Goto[aa]],{k,1,PrimePi[n+4]}];
Print[n," ",0];Label[aa];Continue,{n,1,100}]

A241844 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that L(k) + m is prime, where L(k) is the k-th Lucas number.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 1, 1, 3, 3, 3, 2, 3, 2, 4, 3, 4, 4, 2, 2, 5, 4, 4, 4, 4, 2, 5, 4, 6, 5, 3, 1, 5, 7, 4, 5, 3, 5, 6, 8, 5, 3, 6, 5, 6, 5, 3, 7, 7, 2, 5, 7, 8, 5, 2, 4, 6, 5, 6, 8, 3, 4, 7, 6, 8, 8, 4, 5, 5, 2, 5, 6, 4, 4, 5, 4, 12, 6, 6

Offset: 1

Zhi-Wei Sun, Apr 29 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 1. Moreover, any integer n > 5 can be written as k + m with k > 1 and m > 0 such that L(k) + m is prime.
(ii) Any integer n > 1 can be written as k + m with k > 0 and m > 0 such that F(k) + 2*m is prime, where F(k) is the k-th Fibonacci number given by A000045.
This is similar to the conjecture in A231201. We have verified parts (i) and (ii) for n up to 7*10^6 and 3.7*10^6 respectively.

			a(5) = 1 since 5 = 1 + 4 with L(1) + 4 = 1 + 4 = 5 prime.
a(8) = 1 since 8 = 4 + 4 with L(4) + 4 = 7 + 4 = 11 prime.
a(9) = 1 since 9 = 7 + 2 with L(7) + 2 = 29 + 2 = 31 prime.
a(33) = 1 since 33 = 13 + 20 with L(13) + 20 = 521 + 20 = 541 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000032, A000040, A000045, A000204, A231201.

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

cp's OEIS Frontend

A232109 Least prime p < n + 5 with n + (p-1)*(p-3)/8 prime, or 0 if such a prime p does not exist.

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