A305972 a(n) is the smallest number m such that both |2n-prime(m)| and |2n-prime(m+1)| are primes, where prime(k) is the k-th prime number.

3, 4, 5, 2, 2, 3, 4, 2, 3, 6, 2, 3, 8, 32, 4, 21, 2, 3, 13, 9, 5, 11, 2, 3, 11, 9, 4, 16, 15, 6, 21, 2, 3, 11, 9, 5, 11, 2, 3, 18, 9, 5, 21, 97, 4, 21, 15, 6, 11, 15, 8, 11, 2, 3, 11, 2, 3, 21, 32, 4, 47, 30, 6, 18, 9, 8, 11, 9, 4, 11, 2, 3, 21, 32, 5, 21, 2

Offset

1,1

Comments

In the sequence {a(n)}, the following 30 values of n are the only ones for which prime(a(n)+1) > 2n: 1, 2, 3, 14, 16, 19, 28, 31, 44, 59, 61, 62, 79, 103, 104, 106, 124, 163, 166, 209, 229, 239, 259, 272, 314, 404, 691, 859, 1483, 2011.

Except for the above values of n, the number pairs (prime(a(n)), 2n-prime(a(n))) and (prime(a(n)+1), 2n-prime(a(n)+1)) are the Goldbach decompositions of 2n.

Links

Table of n, a(n) for n=1..77.

Example

For n=1, 2n=2, both |2-5|=3 and |2-7|=5 are primes, and 5 is the 3rd prime, so a(1)=3;
For n=4, 2n=8, both 8-3=5 and 8-5=3 are primes, and 3 is the 2nd prime, so a(4)=2;
For n=13, 2n=26, both 26-prime(8)=26-19=7 and 26-prime(8+1)=26-23=3 are primes, and no smaller primes satisfy the definition, so a(13)=8.

Mathematica

Table[k=1; i=2*n; p=Prime[k]; While[!((PrimeQ[i-p]) && (PrimeQ[i-NextPrime[p]])), k++; p=NextPrime[p]]; If[Prime[k+1]>i, AppendTo[s4, i]]; k, {n, 1, 77}]

Prog

(PARI) a(n) = {my(k=1); while (! (isprime(abs(2*n-prime(k))) && isprime(abs(2*n-prime(k+1)))), k++); k; } \\ Michel Marcus, Jun 22 2018

Crossrefs

Cf. A000040, A002375, A045917, A002372.

Sequence in context: A247369 A009389 A091828 * A121845 A239639 A099816

Adjacent sequences: A305969 A305970 A305971 * A305973 A305974 A305975

Keyword

nonn,easy

Author

Lei Zhou, Jun 15 2018

Status

approved