A057840 Numbers k such that k | 8^k + 6^k + 4^k + 2^k + 1.

1, 3, 7, 9, 27, 49, 81, 133, 243, 267, 343, 729, 2187, 2401, 5999, 6561, 14063, 14337, 16807, 17253, 19683, 22329, 33323, 45619, 59049, 75573, 117649, 144531, 177147, 348519, 383913, 531441, 745339, 823543, 911853, 1594323, 2384883, 3610523, 3814857, 4782969, 5764801

Offset

1,2

Links

Table of n, a(n) for n=1..41.

Mathematica

Select[ Range[ 10^6 ], Mod[ PowerMod[ 8, #, # ] + PowerMod[ 6, #, # ] + PowerMod[ 4, #, # ] + PowerMod[ 2, #, # ] + 1, # ] == 0 & ]

Prog

(PARI) is(k)=sum(i=1, 4, Mod(2*i, k)^k)==-1; \\ Jinyuan Wang, Jun 18 2026
(Python)
from itertools import count, islice
def A057840_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(startvalue, 1)):
        a, b = pow(2, k, k), pow(3, k, k)
        c = ((a+1)*a+b+1)%k
        c = c*a%k
        if c == k-1:
            yield k
A057840_list = list(islice(A057840_gen(), 30)) # Chai Wah Wu, Jun 19 2026

Crossrefs

Cf. A006521, A057839, A057842, A057845.

Sequence in context: A352016 A018827 A249664 * A339614 A246659 A072087

Adjacent sequences: A057837 A057838 A057839 * A057841 A057842 A057843

Keyword

nonn,changed

Author

Robert G. Wilson v, Nov 09 2000

Extensions

a(36)-a(41) from Jinyuan Wang, Jun 18 2026

Status

approved