A252655 Smallest prime p with property that the sum of the n-th power of the successive gaps between primes <= p is also a prime number.

5, 5, 7, 5, 43, 13, 7, 5, 241, 13, 43, 41, 19, 41, 7, 5, 13, 83, 43, 229, 811, 41, 31, 167, 811, 127, 367, 419, 79, 43, 43, 83, 673, 19, 109, 83, 13, 331, 523, 409, 199

Offset: 1

Author

Abhiram R Devesh, Dec 19 2014

Keywords

• nonn

Comments

First appearance of p, by power, beginning with 5: 1, 3, ??, 6, ??, 13, ??, 86, 23, ??, 12, 5, ... . - Robert G. Wilson v, Jan 11 2015

Examples

			n=1: p=5; primes less than or equal to 5: [2, 3, 5]; prime gaps: [1, 2]; sum of prime gaps: 3.
n=2: p=5; primes less than or equal to 5: [2, 3, 5]; squares of prime gaps: [1, 4]; sum of squares of prime gaps: 5.
n=3: p=7; primes less than or equal to 7: [2, 3, 5, 7]; cubes of prime gaps: [1, 8, 8]; sum of cubes of prime gaps: 17.
n=4: p=5; primes less than or equal to 5: [2, 3, 5]; 4th power of prime gaps: [1, 16]; sum of 4th power of prime gaps: 17.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..1000 (first 100 terms from Abhiram R Devesh)

Crossrefs

Cf. A006512, A247177, A247178, A251623.

Programs

Mathematica

f[n_] := Block[{p = 2, s = 0}, While[ !PrimeQ@ s, q = NextPrime@ p; s = s + (q - p)^n; p = q]; p]; Array[f, 60] (* Robert G. Wilson v, Jan 11 2015 *)

Python

import sympy
c=1
while c>0:
    p=2
    d=0
    s=0
    while p>0:
        s=s+(d**c)
        sp=sympy.isprime(s)
        if sp ==True:
            print(c,p)
            p=-1
            c=c+1
        else:
            np=sympy.nextprime(p)
            d=np-p
            p=np