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

%I A251623 #37 May 01 2021 11:41:49
%S A251623 5,19,29,41,61,67,83,89,103,113,167,179,229,263,281,283,307,317,359,
%T A251623 461,467,509,563,571,613,739,743,761,1019,1031,1051,1093,1229,1291,
%U A251623 1297,1319,1409,1447,1609,1621,1667,1747,1801,1877,1979,2113,2137,2161
%N A251623 Primes p with property that the sum of the 4th powers of the successive gaps between primes <= p is a prime number.
%H A251623 Abhiram R Devesh, <a href="/A251623/b251623.txt">Table of n, a(n) for n = 1..1000</a>
%e A251623 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.
%e A251623 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.
%t A251623 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 *)
%o A251623 (Python)
%o A251623 import sympy
%o A251623 p=2
%o A251623 s=0
%o A251623 while 10000>p>0:
%o A251623     np=sympy.nextprime(p)
%o A251623     if sympy.isprime(s):
%o A251623         print(p)
%o A251623     d=np-p
%o A251623     s+=(d**4)
%o A251623     p=np
%o A251623 (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
%Y A251623 Cf. A006512 (with gaps), A247177 (with squares of gaps), A247178 (with cubes of gaps).
%K A251623 nonn,easy
%O A251623 1,1
%A A251623 _Abhiram R Devesh_, Dec 06 2014