A057839 Numbers k such that k | 10^k + 8^k + 6^k + 4^k + 2^k + 1.

1, 31, 198089, 876881261, 2026728077, 24973890553

Offset

1,2

Comments

a(7) > 10^11. - Jason Yuen, Jan 18 2026

Links

Table of n, a(n) for n=1..6.

Mathematica

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

Prog

(Python)
def afind(limit):
  for k in range(1, limit+1, 2):
    if (1 + sum(pow(e, k, k) for e in [2, 4, 6, 8, 10])) % k == 0:
      print(k, end=", ")
afind(10**6) # Michael S. Branicky, Apr 01 2021 after Robert G. Wilson v
(Python)
from itertools import count, islice
def A057839_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(startvalue, 1)):
        a2, a3 = pow(2, k, k), (pow(3, k, k)+pow(5, k, k)+1)%k
        c = ((a2+1)*a2+a3)%k
        c = c*a2%k
        if c == k-1:
            yield k
A057839_list = list(islice(A057839_gen(), 30)) # Chai Wah Wu, Jun 19 2026
(PARI) is_ok(k)=(sum(j=1, 5, Mod(2*j, k)^k) + 1)==0;  \\ Joerg Arndt, Apr 02 2021

Crossrefs

Sequence in context: A351135 A107122 A059113 * A123524 A123523 A079595

Adjacent sequences: A057836 A057837 A057838 * A057840 A057841 A057842

Keyword

nonn,hard,more,changed

Author

Robert G. Wilson v, Nov 09 2000

Extensions

Title corrected by Sean A. Irvine, Mar 31 2021

a(4)-a(5) from Michael S. Branicky, Apr 01 2021

a(6) from Jason Yuen, Jan 18 2026

Status

approved