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Common prime factors of A007663(183)/1093 and A007663(490)/3511.
1

%I #10 Apr 30 2021 12:51:48

%S 3,7,79,2731,8191,121369,22366891

%N Common prime factors of A007663(183)/1093 and A007663(490)/3511.

%C The sequence is complete (cf. Michon).

%C Are these primes necessarily prime factors of A007663(i)/p when p = prime(i) is a Wieferich prime (A001220)?

%C Note the following curious observation: 3 * 7 * 79 * 2731 * 8191 * 121369 * 22366891 = 100743818301219097892181. That number is a repdigit in bases 4 and 64 and undulating in bases 2, 8, 128, 2048 and 8192. Compare that to observations from John Blythe Dobson (cf. Dobson).

%H J. B. Dobson, <a href="https://johnblythedobson.org/mathematics/Wieferich_primes.html">A note on the two known Wieferich primes</a>

%H Gerard P. Michon, <a href="http://www.numericana.com/data/wieferich.htm">Wieferich Primes 1093 and 3511</a>

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

%o my(x=fq(1093)/1093, y=fq(3511)/3511); forprime(p=1, , if(Mod(x, p)==0 && Mod(y, p)==0, print1(p, ", ")))

%Y Cf. A001220, A007663, A010785, A033619, A172290, A242715.

%K nonn,fini,full

%O 1,1

%A _Felix Fröhlich_, Apr 28 2021