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

2, 3, 5, 7, 31, 257, 131071, 618970019642690137449562111, 162259276829213363391578010288127

Offset: 1

Author

Michael De Vlieger, Nov 26 2024

Keywords

• nonn
• hard
• more

Comments

Primes p < 11 are in the sequence since the smallest number in A126706 is 12.

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.

This sequence is that of primes where j = 2^m and k is squarefree.

Proper subset of A141453.

The neighbor k is also divisible by 3, since abs(p-k) = 1 and neither are divisible by 3. Therefore, 6 | k.

Examples

			17 = 2^4+1 is not in the sequence since 18 = 2 * 3^2.
31 = 2^5-1 is in the sequence since 30 = 2*3*5 is squarefree.
127 = 2^7-1 is not in the sequence because 126 = 2 * 3^2 * 7.
257 = 2^8+1 is in the sequence since 258 = 2*3*43 is squarefree.
8191 = 2^13-1 is not in the sequence because 8190 = 2 * 3^2 * 5 * 7 * 13.
65537 = 2^16+1 is not in the sequence since 65538 = 2 * 3^2 * ll * 331.
131071 = 2^17-1 is in the sequence since 131070 = 2 * 3 * 5 * 17 * 257, etc
		

Crossrefs

Cf. A000040, A000079, A126706, A141453, A303554.

Programs

Mathematica

Reap[Do[Which[
  And[PrimeQ[# + 1], SquareFreeQ[(# + 2)/6]], Sow[# + 1],
  And[PrimeQ[# - 1], SquareFreeQ[(# - 2)/6]], Sow[# - 1] ] &[2^i],
{i, 0, 650}] ][[-1, 1]]