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A378108 Primes p such that neither p-1 nor p+1 are in A126706.

%I A378108 #30 Nov 27 2024 18:34:27
%S A378108 2,3,5,7,31,257,131071,618970019642690137449562111,
%T A378108 162259276829213363391578010288127
%N A378108 Primes p such that neither p-1 nor p+1 are in A126706.
%C A378108 Primes p < 11 are in the sequence since the smallest number in A126706 is 12.
%C A378108 Consider p > 11, odd primes; then both p-1 and p+1 are even. Let j and k be neighbors of p. One neighbor, j, is also divisible by 4, while the other neighbor k is not divisible by 2^m, m > 1. The latter statement implies k cannot be a perfect power q^m, p != q, m > 0, but q^m may divide k.
%C A378108 This sequence is that of primes where j = 2^m and k is squarefree.
%C A378108 Proper subset of A141453.
%C A378108 The neighbor k is also divisible by 3, since abs(p-k) = 1 and neither are divisible by 3. Therefore, 6 | k.
%e A378108 17 = 2^4+1 is not in the sequence since 18 = 2 * 3^2.
%e A378108 31 = 2^5-1 is in the sequence since 30 = 2*3*5 is squarefree.
%e A378108 127 = 2^7-1 is not in the sequence because 126 = 2 * 3^2 * 7.
%e A378108 257 = 2^8+1 is in the sequence since 258 = 2*3*43 is squarefree.
%e A378108 8191 = 2^13-1 is not in the sequence because 8190 = 2 * 3^2 * 5 * 7 * 13.
%e A378108 65537 = 2^16+1 is not in the sequence since 65538 = 2 * 3^2 * ll * 331.
%e A378108 131071 = 2^17-1 is in the sequence since 131070 = 2 * 3 * 5 * 17 * 257, etc
%t A378108 Reap[Do[Which[
%t A378108   And[PrimeQ[# + 1], SquareFreeQ[(# + 2)/6]], Sow[# + 1],
%t A378108   And[PrimeQ[# - 1], SquareFreeQ[(# - 2)/6]], Sow[# - 1] ] &[2^i],
%t A378108 {i, 0, 650}] ][[-1, 1]]
%Y A378108 Cf. A000040, A000079, A126706, A141453, A303554.
%K A378108 nonn,hard,more
%O A378108 1,1
%A A378108 _Michael De Vlieger_, Nov 26 2024