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

Author

Felix Fröhlich, Jul 18 2021

Keywords

• nonn

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)

Examples

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

Crossrefs

Cf. A000040, A001220, A007663, A343763.

Programs

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, ", ")))