A389854 Numbers k such that both 2^k-1 and 2^k+1 are squarefree with the same number of distinct prime factors.

2, 11, 14, 23, 29, 47, 53, 71, 73, 74, 82, 86, 95, 101, 113, 115, 121, 142, 167, 169, 179, 181, 199, 203, 209, 233, 235, 277, 307, 311, 317, 335, 337, 343, 347, 349, 353, 355, 358, 361, 382, 434, 449, 494, 509, 515, 518, 529, 535, 547, 583, 599, 613, 634, 643, 653

Offset

1,1

Comments

Numbers k in A262978 such that both omega(A000051(k)) = omega(A000225(k)), where omega = A001221.

Links

Amiram Eldar, Table of n, a(n) for n = 1..89

Example

n  a(n)  s(n)  2^a(n)-1            2^a(n)+1
----------------------------------------------------
1    2     1   3                   5
2   11     2   23 * 89             3 * 683
3   14     3   3 * 43 * 127        5 * 29 * 113
4   23     2   47 * 178481         3 * 2796203
5   29     3   233 * 1103 * 2089   3 * 59 * 3033169

Mathematica

Select[Select[Range[120], AllTrue[2^# + {-1, 1}, SquareFreeQ] &], Apply[PrimeNu[#1] == PrimeNu[#2] &, 2^# + {-1, 1}] &]

Crossrefs

Subsequence of A262978.

Cf. A000051, A000225, A001221, A389853.

Sequence in context: A031192 A283930 A034039 * A077475 A371071 A374446

Adjacent sequences: A389851 A389852 A389853 * A389855 A389856 A389857

Keyword

nonn

Author

Michael De Vlieger, Oct 17 2025

Extensions

More terms from Amiram Eldar, Oct 17 2025

Status

approved