cp's OEIS Frontend

A363544 Least prime p such that 2n can be written as the sum or absolute difference of p and the next prime, or -1 if no such prime exists.

%I A363544 #32 Jul 06 2023 16:57:28
%S A363544 -1,3,7,23,3,139,5,113,1831,7,887,1129,11,2477,2971,13,5591,1327,17,
%T A363544 30593,19333,19,15683,81463,28229,31907,23,35617,82073,44293,29,34061,
%U A363544 89689,162143,31,173359,31397,404597,212701,37,542603,265621,41,155921,544279,43,927869,1100977
%N A363544 Least prime p such that 2n can be written as the sum or absolute difference of p and the next prime, or -1 if no such prime exists.
%C A363544 In recent years, a number of problems have been investigated that concern representing integers as the signed sum of consecutive prime numbers. See, for example, A327467 and the Rivera link.
%C A363544 A000230, which concerns prime gaps, can be considered a more historic such sequence. Here we look at a minor generalization of A000230 in the spirit of signed sums.
%C A363544 When a(n) <> -1, a(n) together with the next prime generate a satisfactory example for proving A362465(2n) = 2.
%H A363544 Carlos Rivera, <a href="https://www.primepuzzles.net/conjectures/conj_021.htm">Conjecture 21. Rivera's conjecture</a>, The Prime Puzzles and Problems Connection.
%F A363544 If 2n is in A001043 then a(n) = prime(k), where k is the position of 2n in A001043, otherwise for n > 0, a(n) = A000230(n).
%F A363544 a(n) = -1 if and only if A362465(2n) <> 2.
%o A363544 (Python)
%o A363544 from sympy import sieve as prime
%o A363544 def A363544(n):
%o A363544     if n == 0: return -1
%o A363544     k = 2
%o A363544     while (prime[k] + prime[k+1]) < 2*n and (prime[k] + prime[k+1]) // 2 != n: k += 1
%o A363544     if (prime[k] + prime[k+1]) // 2 == n: return prime[k]
%o A363544     k = 2
%o A363544     while (prime[k+1] - prime[k]) // 2 != n: k += 1
%o A363544     return prime[k]
%o A363544 print([A363544(n) for n in range(0,50)])
%Y A363544 Cf. A000040, A000230, A001043, A001223, A327467, A359199, A362465.
%K A363544 sign
%O A363544 0,2
%A A363544 _Karl-Heinz Hofmann_ and _Peter Munn_, Jun 09 2023