A346431 Primes p such that A007663(i) is divisible by Product_{k=1..7} A343763(k), where i is the index of p in A000040.

157, 313, 547, 859, 937, 1093, 1171, 1249, 1327, 1483, 1873, 1951, 2029, 2341, 2887, 3121, 3433, 3511, 3823, 4057, 4447, 4603, 4759, 4993, 5227, 5851, 6007, 6163, 6397, 6553, 6709, 7177, 7333, 7411, 7489, 7723, 7879, 8269, 8581, 8737, 8893, 8971, 9049, 9127

Offset

1,1

Comments

Differs from A142159 in that 79, 2731, 8191, ... are not in this sequence.

Includes the two known Wieferich primes 1093 and 3511 (cf. A001220).

Is this a supersequence of A001220, i.e., are all Wieferich primes in the sequence?

Is p-1 always divisible by 78 = 2 * 3 * 13?

For the initial primes p in this sequence, p-1 has some interesting digit patterns in various bases, as illustrated in the following table:

p | b | base-b expansion of p-1

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

157 | 5 | 1111

313 | 5 | 2222

547 | 3 | 202020

547 | 4 | 20202

547 | 5 | 4141

547 | 9 | 666

547 | 16 | 222

859 | 2 | 1101011010

937 | 3 | 1021200 (nearly palindromic)

937 | 4 | 32220 (nearly palindromic)

937 | 5 | 12221

1093 | 2 | 10001000100 (periodic)

1093 | 3 | 1111110 (nearly palindromic/repdigit)

1093 | 4 | 101010

1093 | 5 | 13332 (nearly palindromic)

1093 | 16 | 444

1171 | 2 | 10010010010 (periodic)

1171 | 5 | 14140 (nearly palindromic and periodic)

1171 | 8 | 2222

1249 | 3 | 1201020 (nearly palindromic)

1249 | 5 | 14443 (nearly palindromic)

1327 | 5 | 20301 (nearly palindromic)

Links

Table of n, a(n) for n=1..44.

Example

(2^(157-1)-1)/157 is divisible by 3 * 7 * 79 * 2731 * 8191 * 121369 * 22366891, so 157 is a term of the sequence.

Prog

(PARI) fq(n) = (2^(n-1)-1)/n
my(prd=3*7*79*2731*8191*121369*22366891); forprime(p=1, , if(Mod(fq(p), prd)==0, print1(p, ", ")))

Crossrefs

Cf. A000040, A001220, A007663, A343763.

Sequence in context: A142581 A140625 A142874 * A234364 A060974 A073277

Adjacent sequences: A346428 A346429 A346430 * A346432 A346433 A346434

Keyword

nonn

Author

Felix Fröhlich, Jul 18 2021

Status

approved