cp's OEIS Frontend

A277854 Frequent terms, i.e., values such that no smaller value appears more often, in A075771 = quotient + remainder of Euclidean division of n^2 by prime(n).

%I A277854 #12 Oct 10 2017 22:20:33
%S A277854 1,2,4,5,9,15,16,48,64,86,100,144,169,3364,3969,4096,195364
%N A277854 Frequent terms, i.e., values such that no smaller value appears more often, in A075771 = quotient + remainder of Euclidean division of n^2 by prime(n).
%C A277854 Equivalently, record values (in the weak sense of >=) in the sequence of frequencies of values of A075771. (The lower bound A075771(n) >= n^2/prime(n) ensures that no number below this limit can occur beyond the index n in that sequence.)
%C A277854 It appears that this sequence contains mainly squares, but there are exceptions such as 2, 5, 15, 48, 86, and some squares (25 = 5^2, 36 = 6^2, 49 = 7^2, 81 = 9^2, 121 = 11^2) do not occur. Is there an explanation for this and/or the fact that exceptions are close to missing squares: 48 ~ 49 = 7^2, 86 ~ 81 = 9^2 ? Can one prove or disprove that
%C A277854 - from some point on, only squares will occur?
%C A277854 - all sufficiently large squares (or: even squares?) will occur?
%C A277854 - from a(12) = 144 (or some later point) on, a(n) will occur in A075771 strictly more often than the preceding value?
%C A277854 a(18) > 10^6. - _Robert G. Wilson v_, Nov 25 2016
%e A277854 Values that occur in A075771 not less often than any smaller value are 1, 2, 4 (which appear once), 5, 9, 15 (which appear twice), 16, 48, 64, 86, 100 (which appear three times), 144 (which appears five times), 169 (which appears seven times), ...
%Y A277854 Cf. A075771, A277851, A277852, A277853.
%K A277854 nonn,more
%O A277854 1,2
%A A277854 _M. F. Hasler_, Nov 25 2016
%E A277854 a(14)-a(17) from _Robert G. Wilson v_, Nov 25 2016