A280223 Distance of n to the next number with middle divisors.

1, 2, 1, 2, 1, 2, 1, 1, 3, 2, 1, 3, 2, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 1, 3, 2, 1, 2, 1, 2, 1, 3, 2, 1, 1, 4, 3, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1, 1, 1, 4, 3, 2, 1, 2, 1, 4, 3, 2, 1, 3, 2, 1, 1, 2, 1, 4, 3, 2, 1, 2, 1, 5, 4, 3, 2, 1, 3, 2, 1, 1, 3, 2, 1, 4, 3, 2, 1, 2, 1, 1, 5, 4, 3, 2, 1, 2, 1, 1, 1, 4, 3, 2, 1, 4, 3, 2, 1

Offset

1,2

Comments

The original name was: Precipice of n: descending by the main diagonal of the pyramid described in A245092, a(n) is the height difference between the n-th level (starting from the top) and the level of the next terrace.

The structure of the stepped pyramid arises after the 90-degree-zig-zag folding of the diagram of the isosceles triangle A237593.

The terraces at the n-th level of the pyramid are also the parts of the symmetric representation of sigma(n).

The stepped pyramid is also one of the 3D-quadrants of the stepped pyramid described in A244050.

Note that if a(n) > 1 then the next k terms are the first k positive integers in decreasing order, where k = a(n) - 1.

For more information about the precipices see A277437 and A280295.

a(n) is also the number of numbers >= n whose largest Dyck paths of the symmetric representation of sigma share the same point at the main diagonal of the diagram. For more information see A237593.

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Omar E. Pol, Perspective view of the pyramid (first 16 levels)

Example

Descending by the main diagonal of the stepped pyramid, for the levels 9, 10 and 11 we have that the next terrace is in the 12th level, so a(9) = 12 - 9 = 3, a(10) = 12 - 10 = 2, and a(11) = 12 - 11 = 1.

Mathematica

A280223[n_] := Module[{k = n, min, max}, While[min = Sqrt[++k/2]; max = Sqrt[2*k]; NoneTrue[Divisors[k], min <= # < max &]]; k - n];
Array[A280223, 100] (* Paolo Xausa, Nov 07 2025 *)

Crossrefs

Cf. A000203, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A240542, A244050, A245092, A259179, A262626, A277437, A279286, A279385, A280295.

Cf. A067742, A071562, A390114.

Sequence in context: A343854 A069349 A167404 * A062754 A342846 A045888

Adjacent sequences: A280220 A280221 A280222 * A280224 A280225 A280226

Keyword

nonn

Author

Omar E. Pol, Dec 29 2016

Extensions

More terms from Omar E. Pol, Jan 02 2017

New name from Omar E. Pol, Nov 02 2025

Status

approved