A231725 Least positive integer k < n such that n + k + 2^k is prime, or 0 if such an integer k does not exist.

0, 1, 0, 1, 2, 3, 2, 1, 4, 1, 2, 3, 2, 1, 10, 1, 2, 3, 6, 1, 4, 5, 2, 5, 2, 1, 4, 1, 8, 3, 2, 3, 4, 1, 2, 3, 2, 1, 4, 1, 2, 3, 6, 1, 12, 5, 2, 3, 8, 1, 4, 5, 2, 11, 2, 1, 6, 1, 4, 3, 2, 3, 4, 1, 2, 5, 2, 1, 4, 1, 22, 3, 2, 57, 10, 1, 2, 3, 6, 1, 4, 11, 2, 11, 8, 1, 4, 7, 4, 3, 2, 3, 4, 1, 2, 3, 2, 1, 16, 1

Offset

1,5

Comments

This was motivated by A231201 and A231557.

Conjecture: a(n) > 0 for all n > 3. We have verified this for n up to 2*10^6; for example, we find the following relatively large values of a(n): a(65958) = 37055, a(299591) = 51116, a(295975) = 13128, a(657671) = 25724, a(797083) = 44940, a(1278071) = 24146, a(1299037) = 34502, a(1351668) = 25121, a(1607237) = 34606, a(1710792) = 11187, a(1712889) = 18438.

I conjecture the opposite. In particular I expect that a(n) = 0 for infinitely many values of n. - Charles R Greathouse IV, Nov 13 2013

Links

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.

Example

a(3) = 0 since 3 + 1 + 2^1 = 6 and 3 + 2 + 2^2 = 9 are both composite.
a(5) = 2 since 5 + 1 + 2^1 = 8 is not prime, but 5 + 2 + 2^2 = 11 is prime.

Mathematica

Do[Do[If[PrimeQ[n+k+2^k], Print[n, " ", k]; Goto[aa]], {k, 1, n-1}];
Print[n, " ", 0]; Label[aa]; Continue, {n, 1, 100}]

Prog

(PARI) a(n)=for(k=1, n-1, if(ispseudoprime(n+k+2^k), return(k))); 0 \\ Charles R Greathouse IV, Nov 13 2013

Crossrefs

Cf. A000040, A000079, A231201, A231516, A231557, A231561, A231631, A231776.

Sequence in context: A168069 A334014 A280929 * A106559 A280047 A106377

Adjacent sequences: A231722 A231723 A231724 * A231726 A231727 A231728

Keyword

nonn

Author

Zhi-Wei Sun, Nov 12 2013

Status

approved