A251623 Primes p with property that the sum of the 4th powers of the successive gaps between primes <= p is a prime number.

5, 19, 29, 41, 61, 67, 83, 89, 103, 113, 167, 179, 229, 263, 281, 283, 307, 317, 359, 461, 467, 509, 563, 571, 613, 739, 743, 761, 1019, 1031, 1051, 1093, 1229, 1291, 1297, 1319, 1409, 1447, 1609, 1621, 1667, 1747, 1801, 1877, 1979, 2113, 2137, 2161

Offset: 1

Author

Abhiram R Devesh, Dec 06 2014

Keywords

• nonn
• easy

Examples

			a(1)=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.
a(2)=19; primes less than or equal to 13: [2, 3, 5, 7, 11, 13, 17, 19]; 4th powers of prime gaps (see A140299): [1, 16, 16, 256, 16, 256, 16]; sum of these: 577.
		

Links

• Abhiram R Devesh, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A006512 (with gaps), A247177 (with squares of gaps), A247178 (with cubes of gaps).

Programs

Mathematica

p = 2; q = 3; s = 0; lst = {}; While[p < 2500, s = s + (q - p)^4; If[ PrimeQ@ s, AppendTo[lst, q]]; p = q; q = NextPrime@ q]; lst (* Robert G. Wilson v, Dec 19 2014 *)

PARI

p = 2; q = 3; s = 1; for (i = 1, 100, p = q; q = nextprime (q + 1); if (isprime (s = s + (q - p)^4), print1 (q ", "))) \\ Zak Seidov, Jan 19 2015

Python

import sympy
p=2
s=0
while 10000>p>0:
    np=sympy.nextprime(p)
    if sympy.isprime(s):
        print(p)
    d=np-p
    s+=(d**4)
    p=np