cp's OEIS Frontend

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.

%I A252655 #36 May 22 2025 10:21:41
%S A252655 5,5,7,5,43,13,7,5,241,13,43,41,19,41,7,5,13,83,43,229,811,41,31,167,
%T A252655 811,127,367,419,79,43,43,83,673,19,109,83,13,331,523,409,199
%N 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.
%C A252655 First appearance of p, by power, beginning with 5: 1, 3, ??, 6, ??, 13, ??, 86, 23, ??, 12, 5, ... . - _Robert G. Wilson v_, Jan 11 2015
%H A252655 Robert G. Wilson v, <a href="/A252655/b252655.txt">Table of n, a(n) for n = 1..1000</a> (first 100 terms from Abhiram R Devesh)
%e A252655 n=1: p=5; primes less than or equal to 5: [2, 3, 5]; prime gaps: [1, 2]; sum of prime gaps: 3.
%e A252655 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.
%e A252655 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.
%e A252655 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.
%t A252655 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 *)
%o A252655 (Python)
%o A252655 import sympy
%o A252655 c=1
%o A252655 while c>0:
%o A252655     p=2
%o A252655     d=0
%o A252655     s=0
%o A252655     while p>0:
%o A252655         s=s+(d**c)
%o A252655         sp=sympy.isprime(s)
%o A252655         if sp ==True:
%o A252655             print(c,p)
%o A252655             p=-1
%o A252655             c=c+1
%o A252655         else:
%o A252655             np=sympy.nextprime(p)
%o A252655             d=np-p
%o A252655             p=np
%Y A252655 Cf. A006512, A247177, A247178, A251623.
%K A252655 nonn
%O A252655 1,1
%A A252655 _Abhiram R Devesh_, Dec 19 2014