
COMMENTS

For j >= 3, 1 + 5*10^j = A199685(j) is in the sequence, so the sequence is infinite.  Vaclav Kotesovec, Mar 29 2018
From Robert Dawson, Apr 12 2018: (Start)
This sequence is the union of the following ten subsequences.
Terms in <angle brackets> have fewer than d digits: they are always terms of the sequence, and always appear elsewhere, as an earlier term of the same subsequence or a related subsequence. (However, the dth terms of the subsequences are always distinct for any d > 4.) Dashes replace certain solutions to the congruences for small values of d for which certain other divisibility criteria are not met. The integers n_0(d) and n_1(d) are the even and odd zeros of n^2+3n+4 (mod 2^d) (note that by Hensel's Lemma these always exist and each is unique).
(i) p(d) satisfying 2^d p(d)  n_0(d), 5^d p(d):
(0,<0>,500,2500,62500,62500,4062500,14062500,...)
(ii) q(d) satisfying 2^{d1}q(d)1, 5^dq(d) for d != 3:
(0,25,,<625>,40625,390625,2890625,12890625,...)
(iii) q(d) + 5x10^{d1} for d != 2:
(5,, 625,5625,90625, 890625,7890625, 62890625,...)
(iv) q'(d) satisfying 2^{d1}q'(d)  n_1(d), 5^dq'(d), for d != 1,3:
(,25,,<625>,15625,265625,2265625,47265625,...)
(v) q'(d) + 5x10^{d1} for d != 2:
(5,,625,5625,65625,765625,7265625,97265625,...)
(vi) r(d) satisfying 2^dr(d), 5^dr(d)1 for d >= 2
(,76,376,9376,<9376>,109376,7109376,87109376,...) = A016090(d)
(vii) r'(d) satisfying 2^dr'(d)  n_0(d), 5^dr'(d)1 for d >= 2:
(,76,876,1876,71876,171876,1171876,<1171876>,...)
(viii)s(d) := 5x10^{d1}+1 for d >= 4:
(,,,5001,50001,500001,5000001,50000001,...) = A199685(d1)
(ix) t(d) satisfying 2^{d1}t(d)n_0(d), 5^dt(d)1:
(1,<1>,<1>,<1>,25001,375001,4375001,34375001,...)
(x) t(d) + 5x10^{d1} for d >= 4:
(,,,5001,75001,875001,9375001,84375001,...)
For d > 4, the sequence A301912 has at most 10 and at least 5 terms with d digits. The maximum is first attained for d=7. The minimum is first attained for d=168.
(End.)
