A352217 Smallest power of 2 that is one more than a multiple of 2n-1.

2, 4, 16, 8, 64, 1024, 4096, 16, 256, 262144, 64, 2048, 1048576, 262144, 268435456, 32, 1024, 4096, 68719476736, 4096, 1048576, 16384, 4096, 8388608, 2097152, 256, 4503599627370496, 1048576, 262144, 288230376151711744, 1152921504606846976, 64, 4096

Offset: 1

Author

J. Lowell, Mar 07 2022

Keywords

• nonn
• easy

Comments

Every odd number is a divisor of a number of the form 2^n-1.

Examples

			a(5)=64 because 63 is the smallest number of the form 2^n-1 that's a multiple of 9.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..1661

Crossrefs

Cf. A002326, A005408, A165781.

Programs

Maple

a:= n-> 2^`if`(n=1, 1, numtheory[order](2, 2*n-1)):
seq(a(n), n=1..50);  # Alois P. Heinz, Mar 07 2022

Mathematica

Table[2^MultiplicativeOrder[2, 2*n - 1], {n, 1, 33}] (* Amiram Eldar, Mar 08 2022 *)

PARI

a(n) = 1 << znorder(Mod(2,2*n-1)); \\ Kevin Ryde, Mar 07 2022

Python

def a(n):
    if n == 1: return 2
    p, m = 2, 2*n-1
    while p <= m or p % m != 1: p *= 2
    return p
print([a(n) for n in range(1, 34)]) # Michael S. Branicky, Mar 07 2022

Python

from sympy import n_order
def a(n): return 2**n_order(2, 2*n-1)
print([a(n) for n in range(1, 34)]) # Michael S. Branicky, Mar 07 2022 after Alois P. Heinz

Formula

From Alois P. Heinz, Mar 07 2022: (Start)

a(n) = 2^A002326(n-1).

a(n) = 1 + A165781(n-1)*(2*n-1). (End)

Extensions

a(14) and beyond from Michael S. Branicky, Mar 07 2022