A281728 - 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&amp;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