Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #26 Mar 16 2024 04:35:12
%S 3,11,13,43,241,683,2731,43691,61681,174763,2796203,15790321,
%T 715827883,4278255361,2932031007403,4363953127297
%N Johnson's non-Wieferich numbers of the first kind.
%C This is the case a = 2 of primes p such that p-1 has the a-adic expansion bb...b00...0bb...b00...0_a, where b = a-1 with each of the t blocks of digits b or 0 having length k and additionally q_a == (a^k + 1)/(t + 1)*k =/= 0 (mod p), where q_a denotes the Fermat quotient to base a (cf. Johnson, 1977).
%C These are prime numbers of the form (2^m + 1)/(2^n + 1). Note that if m,n > 0, then 2^n + 1 divides 2^m + 1 if and only if m/n is odd. - _Thomas Ordowski_, Feb 17 2024
%H J. B. Dobson, <a href="https://johnblythedobson.org/mathematics/Wieferich_primes.html">A note on the two known Wieferich Primes</a> (see section "Johnson's non-Wieferich numbers of the first kind (his Corollary 5 with a = 2, b = 1)").
%H Wells Johnson, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002193698&physid=PHYS_0203">On the nonvanishing of Fermat quotients (mod p)</a>, Journal f. die reine und angewandte Mathematik 292, (1977): 196-200.
%e (2^49+1)/(2^7+1) = 4363953127297 = 111111100000001111111000000011111110000001.
%Y Cf. A370425 (integers of the form (2^m+1)/(2^n+1)).
%K nonn,more
%O 1,1
%A _Felix Fröhlich_, Jan 28 2017