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%I A346431 #10 Jul 21 2021 09:34:42
%S A346431 157,313,547,859,937,1093,1171,1249,1327,1483,1873,1951,2029,2341,
%T A346431 2887,3121,3433,3511,3823,4057,4447,4603,4759,4993,5227,5851,6007,
%U A346431 6163,6397,6553,6709,7177,7333,7411,7489,7723,7879,8269,8581,8737,8893,8971,9049,9127
%N 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.
%C A346431 Differs from A142159 in that 79, 2731, 8191, ... are not in this sequence.
%C A346431 Includes the two known Wieferich primes 1093 and 3511 (cf. A001220).
%C A346431 Is this a supersequence of A001220, i.e., are all Wieferich primes in the sequence?
%C A346431 Is p-1 always divisible by 78 = 2 * 3 * 13?
%C A346431 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 A346431      p |  b | base-b expansion of p-1
%C A346431 --------------------------------------
%C A346431    157 |  5 | 1111
%C A346431    313 |  5 | 2222
%C A346431    547 |  3 | 202020
%C A346431    547 |  4 | 20202
%C A346431    547 |  5 | 4141
%C A346431    547 |  9 | 666
%C A346431    547 | 16 | 222
%C A346431    859 |  2 | 1101011010
%C A346431    937 |  3 | 1021200 (nearly palindromic)
%C A346431    937 |  4 | 32220 (nearly palindromic)
%C A346431    937 |  5 | 12221
%C A346431   1093 |  2 | 10001000100 (periodic)
%C A346431   1093 |  3 | 1111110 (nearly palindromic/repdigit)
%C A346431   1093 |  4 | 101010
%C A346431   1093 |  5 | 13332 (nearly palindromic)
%C A346431   1093 | 16 | 444
%C A346431   1171 |  2 | 10010010010 (periodic)
%C A346431   1171 |  5 | 14140 (nearly palindromic and periodic)
%C A346431   1171 |  8 | 2222
%C A346431   1249 |  3 | 1201020 (nearly palindromic)
%C A346431   1249 |  5 | 14443 (nearly palindromic)
%C A346431   1327 |  5 | 20301 (nearly palindromic)
%e A346431 (2^(157-1)-1)/157 is divisible by 3 * 7 * 79 * 2731 * 8191 * 121369 * 22366891, so 157 is a term of the sequence.
%o A346431 (PARI) fq(n) = (2^(n-1)-1)/n
%o A346431 my(prd=3*7*79*2731*8191*121369*22366891); forprime(p=1, , if(Mod(fq(p), prd)==0, print1(p, ", ")))
%Y A346431 Cf. A000040, A001220, A007663, A343763.
%K A346431 nonn
%O A346431 1,1
%A A346431 _Felix Fröhlich_, Jul 18 2021

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