A277851 -id:A277851 - cp's OEIS frontend

A277852 "Late birds" (values a(n) < a(k) for all k > n) in A075771 = quotient + remainder of Euclidean division of n^2 by prime(n).

1, 2, 4, 5, 9, 15, 16, 24, 59, 61, 81, 100, 124, 129, 152, 169, 196, 249, 305, 400, 425, 475, 520, 556, 565, 676, 771, 795, 904, 939

Offset: 1

M. F. Hasler, Nov 25 2016

The lower bound A075771(n) >= n^2/prime(n) ensures that a given number can't occur beyond a certain index in that sequence.

See A277853 for the corresponding indices m.

Cf. A075771, A277851, A277853, A277854.

A277852(N,L=N^2/prime(N),A=A075771,S=List())={forstep(n=N,1,-1,A(n)

a(n) = A075771(A277853(n)), i.e., equals A075771 o A277853.

A277853 Indices m of "late birds", i.e., values a(m) < a(k) for all k > m, in A075771 = quotient + remainder in Euclidean division of n^2 by prime(n).

Original entry on oeis.org

1, 2, 4, 5, 21, 27, 44, 104, 173, 365, 369, 500, 590, 735, 840, 987, 1564, 1797, 2415, 3368, 3545, 4025, 4466, 5018, 5477, 6686, 7239, 8025, 8182, 9369

Offset: 1

M. F. Hasler, Nov 25 2016

See A277852 (= A075771 o A277853) for the corresponding values of the "late birds".

Cf. A075771, A277851, A277852, A277854.

A277853(N,L=N^2/prime(N),A=A075771,S=List())={forstep(n=N,1,-1,A(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).

Original entry on oeis.org

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

Offset: 1

M. F. Hasler, Nov 25 2016

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

			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), ...

Cf. A075771, A277851, A277852, A277853.

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

cp's OEIS Frontend

A277852 "Late birds" (values a(n) < a(k) for all k > n) in A075771 = quotient + remainder of Euclidean division of n^2 by prime(n).

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A277853 Indices m of "late birds", i.e., values a(m) < a(k) for all k > m, in A075771 = quotient + remainder in Euclidean division of n^2 by prime(n).

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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).

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