A249112 Second smallest k > 0 such that n+(1+2+...+k) is prime.

3, 2, 7, 2, 8, 10, 4, 5, 7, 2, 8, 10, 4, 5, 16, 2, 8, 10, 7, 6, 16, 5, 8, 22, 7, 5, 16, 2, 15, 22, 4, 6, 7, 9, 8, 13, 4, 5, 19, 2, 11, 10, 7, 5, 16, 5, 8, 13, 12, 6, 7, 5, 8, 22, 7, 5, 16, 2, 15, 13, 4, 9, 16, 5, 8, 13, 8, 5, 7, 2, 11, 10, 4, 14, 16, 6, 8

Offset: 1

Author

Gil Broussard, Oct 21 2014

Keywords

• easy
• nonn

Comments

Take the counting numbers and continue adding 1, 2, ..., a(n) until reaching a second prime.

Conjecturally (Hardy & Littlewood conjecture F), a(n) exists for all n. - Charles R Greathouse IV, Oct 21 2014

It appears that the minimum value reached by a(n) is 2, and this occurs for n= 2, 4, 10, 16, 28, 40, 58, 70, ... Is this A144834? - Michel Marcus, Oct 26 2014

Examples

			a(3)=7 because 3+1+2+3+4+5+6+7=31 and one partial sum is prime.
a(4)=2 because 4+1=5 and 4+1+2=7.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A085415, A249113.

Programs

Mathematica

Table[k = 0; Do[k++; While[! PrimeQ[n + Total@ Range@ k], k++], {x, 2}]; k, {n, 77}] (* Michael De Vlieger, Jan 03 2016 *)

PARI

a(n)=my(k, s=2); while(s, if(isprime(n+=k++), s--)); k \\ Charles R Greathouse IV, Oct 21 2014

PARI

a(n,s=2)=my(k);until(isprime(n+=k++)&&!s--,);k \\ allows one to get A249113(n) as a(n,3). - M. F. Hasler, Oct 21 2014

Formula

n+A000217(k) is prime for k=a(n) and exactly one smaller positive value. - M. F. Hasler, Oct 21 2014