A370879 - OEIS

A370879

a(n) = 2^n*t + 1 where t is the least x such that there exists an r in the range 2 <= r <= x+1 that is coprime to 2^n*x + 1 and has multiplicative order 2^n modulo 2^n*x + 1.

3, 5, 17, 193, 353, 641, 769, 10753, 10753, 13313, 12289, 114689, 114689, 163841, 786433, 786433, 6684673, 13631489, 5767169, 7340033, 111149057, 104857601, 167772161, 167772161, 469762049, 2483027969, 2281701377, 3221225473, 12348030977, 52613349377

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

r^(2^(n-1)) == -1 (mod a(n)).

Is there a proof that a(n) always exists?

LINKS

Table of n, a(n) for n=1..30.

EXAMPLE

n | t | r | 2^n*t+1

---------------------

1 | 1 | 2 | 3

2 | 1 | 2 | 5

3 | 2 | 2 | 17

4 | 12 | 3 | 193

5 | 11 | 6 | 353

PROG

(Python)

def a(n):

t = 1

while True:

a_n = (t << n) + 1

for r in range(2, t+2):

if pow(r, 1 << (n-1), a_n) + 1 == a_n:

return a_n

t += 1

print([a(n) for n in range(1, 31)])

(PARI) isok(t, n) = for (r=2, t+1, if ((gcd(r, 2^n*t + 1)==1) && znorder(Mod(r, 2^n*t + 1)) == 2^n, return(1))); return(0);

a(n) = my(t=1); while (!isok(t, n), t++); 2^n*t + 1; \\ Michel Marcus, Mar 17 2024

CROSSREFS

Cf. A035089.

Sequence in context: A083213 A171271 A056826 * A278138 A273870 A272060

Adjacent sequences: A370876 A370877 A370878 * A370880 A370881 A370882

KEYWORD

nonn

AUTHOR

Darío Clavijo, Mar 03 2024

STATUS

approved