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).

1, 2, 4, 5, 9, 15, 16, 48, 64, 86, 100, 144, 169, 3364, 3969, 4096, 195364

Offset: 1

Author

M. F. Hasler, Nov 25 2016

Keywords

• nonn
• more

Comments

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.)

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

- from some point on, only squares will occur?

- all sufficiently large squares (or: even squares?) will occur?

- from a(12) = 144 (or some later point) on, a(n) will occur in A075771 strictly more often than the preceding value?

a(18) > 10^6. - Robert G. Wilson v, Nov 25 2016

Examples

			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), ...
		

Crossrefs

Cf. A075771, A277851, A277852, A277853.

Extensions

a(14)-a(17) from Robert G. Wilson v, Nov 25 2016