Primes p such that
A258367(i) < 2, where i is the index of p in
A000040.
Wieferich primes together with the "closest" near-misses possible that are not actually Wieferich.
Countless such sequences with larger bounds on the value of abs(A) are possible. This is one of the few that I believe should be in the OEIS.
The corresponding sequence of A-values is 1, 1, 1, 1, -1, 0, -1, 0, -1, 1, 1.
I checked the range 3 <= p <= 47004625957 with PARI. 76843523891 is from Crandall, Dilcher, Pomerance, 1997.
There are no near-Wieferich primes with abs(A) < 2 in the range 4*10^12 to 1.25*10^15 (cf. Knauer, Richstein, 2005).
There are no near-Wieferich primes with abs(A) < 2 in the range 1*10^15 to 3*10^15 per information I received from Mark Rodenkirch in 2010.
There are no near-Wieferich primes with abs(A) < 2 in the range 3*10^15 to ~6*10^17 (cf. Goetz, cf. Reggie, cf. Fries).
As of Apr 26 2022, a(12) > ~1.1*10^19 if it exists (cf. WW Statistics).
Heuristically, one would expect about 11 to 12 (3*log(log(10^19))) near-Wieferich primes with |A| <= 1 up to 10^19, a very close match to the actual number of 11 (cf. Crandall, Dilcher, Pomerance, 1997, p. 446).
