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A358127 a(n) is the cardinality of the set of pairwise gcd's of {prime(1)+1, ..., prime(n)+1}.

%I A358127 #14 Nov 02 2022 11:53:24
%S A358127 1,3,4,5,5,5,5,7,8,8,8,9,9,11,12,14,14,14,14,14,14,15,15,15,16,16,18,
%T A358127 19,20,21,22,22,23,23,23,23,23,24,24,26,27,29,29,30,32,32,33,35,36,36,
%U A358127 37,37,37,37,38,38,39,39,39,39,40,40,42,42,43,43,43,44,45,45,48,48,48,48,50,50,50,50
%N A358127 a(n) is the cardinality of the set of pairwise gcd's of {prime(1)+1, ..., prime(n)+1}.
%e A358127 For n = 3 initial set is {2+1, 3+1, 5+1} and after applying gcd for each distinct pair of elements we get {1, 2, 3} set with cardinality of a(3) = 3.
%o A358127 (Python)
%o A358127 from sympy import nextprime
%o A358127 from math import gcd
%o A358127 from itertools import combinations
%o A358127 pr, terms = [2,3], []
%o A358127 for i in range(100):
%o A358127     terms.append(len(set([gcd(t[0]+1, t[1]+1) for t in combinations(pr,2)])))
%o A358127     pr.append(nextprime(pr[-1]))
%o A358127 print(terms)
%o A358127 (Python)
%o A358127 from math import gcd
%o A358127 from itertools import count, islice
%o A358127 from sympy import prime
%o A358127 def A358127_gen(): # generator of terms
%o A358127     a, b = [3], set()
%o A358127     for n in count(2):
%o A358127         q = prime(n)+1
%o A358127         b |= set(gcd(p,q) for p in a)
%o A358127         yield len(b)
%o A358127         a.append(q)
%o A358127 A358127_list = list(islice(A358127_gen(),100)) # _Chai Wah Wu_, Nov 02 2022
%Y A358127 Cf. A008864, A356371, A214799.
%K A358127 nonn
%O A358127 2,2
%A A358127 _Gleb Ivanov_, Oct 30 2022