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Primes p such that A007663(i) is divisible by Product_{k=1..7} A343763(k), where i is the index of p in A000040.
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%I #10 Jul 21 2021 09:34:42

%S 157,313,547,859,937,1093,1171,1249,1327,1483,1873,1951,2029,2341,

%T 2887,3121,3433,3511,3823,4057,4447,4603,4759,4993,5227,5851,6007,

%U 6163,6397,6553,6709,7177,7333,7411,7489,7723,7879,8269,8581,8737,8893,8971,9049,9127

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

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

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

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

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

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

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

%C --------------------------------------

%C 157 | 5 | 1111

%C 313 | 5 | 2222

%C 547 | 3 | 202020

%C 547 | 4 | 20202

%C 547 | 5 | 4141

%C 547 | 9 | 666

%C 547 | 16 | 222

%C 859 | 2 | 1101011010

%C 937 | 3 | 1021200 (nearly palindromic)

%C 937 | 4 | 32220 (nearly palindromic)

%C 937 | 5 | 12221

%C 1093 | 2 | 10001000100 (periodic)

%C 1093 | 3 | 1111110 (nearly palindromic/repdigit)

%C 1093 | 4 | 101010

%C 1093 | 5 | 13332 (nearly palindromic)

%C 1093 | 16 | 444

%C 1171 | 2 | 10010010010 (periodic)

%C 1171 | 5 | 14140 (nearly palindromic and periodic)

%C 1171 | 8 | 2222

%C 1249 | 3 | 1201020 (nearly palindromic)

%C 1249 | 5 | 14443 (nearly palindromic)

%C 1327 | 5 | 20301 (nearly palindromic)

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

%o (PARI) fq(n) = (2^(n-1)-1)/n

%o my(prd=3*7*79*2731*8191*121369*22366891); forprime(p=1, , if(Mod(fq(p), prd)==0, print1(p, ", ")))

%Y Cf. A000040, A001220, A007663, A343763.

%K nonn

%O 1,1

%A _Felix Fröhlich_, Jul 18 2021