A245092 -id:A245092 - cp's OEIS frontend

A280919 Precipices from the successive terraces, descending by the main diagonal of the pyramid described in A245092. Also first differences of A071562.

1, 2, 2, 2, 1, 3, 3, 1, 2, 2, 4, 1, 3, 2, 2, 3, 1, 4, 2, 3, 3, 1, 1, 4, 2, 4, 3, 1, 2, 4, 2, 5, 3, 1, 3, 4, 2, 1, 5, 2, 1, 1, 4, 4, 2, 2, 5, 3, 1, 5, 2, 2, 2, 3, 5, 3, 1, 6, 3, 1, 2, 4, 2, 3, 3, 1, 1, 6, 4, 2, 5, 3, 2, 3, 1, 2, 2, 4, 4, 1, 1, 6, 4, 1, 3, 1, 3

Offset: 1

Omar E. Pol, Jan 10 2017

nonn

Descending by the main diagonal of the pyramid, A071562 gives the levels where we can find a terrace.
The terraces at the k-th level of the pyramid are also the parts of the symmetric representation of sigma(k).
a(n) is the length of the n-th vertical line segment at the main diagonal of the pyramid.
a(n) is the precipice of A071562(n).
The structure of the stepped pyramid arises after the 90-degree-zig-zag folding of the diagram of the isosceles triangle A237593.
The stepped pyramid is also one of the 3D-quadrants of the stepped pyramid described in A244050.
Equals nonzero terms of A259179. - Omar E. Pol, Apr 17 2018

Robert Price, Table of n, a(n) for n = 1..13750
Omar E. Pol, Illustration of the isosceles triangle of A237593 (rows 1..28)
Omar E. Pol, Perspective view of the pyramid (first 16 levels)

For more information about the precipices see A276112, A277437, A280223 and A280295.

Cf. A000203, A071562, A196020, A235791, A236104, A237048, A237591, A237593, A237270, A237271, A244050, A245092, A259179, A262626.

Differences@ Select[Range@ 228, Function[n, Total@ Select[Divisors@ n, Sqrt[n/2] <= # < Sqrt[2 n] &] != 0]] (* Michael De Vlieger, Jan 13 2017, after Robert G. Wilson v at A071562 *)

a(n) = A280223(A071562(n)).

More terms from Michael De Vlieger, Jan 13 2017

A280223 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.

Original entry on oeis.org

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

Omar E. Pol, Dec 29 2016

nonn

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.

			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.

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

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

More terms from Omar E. Pol, Jan 02 2017

A277437 Square array read by antidiagonals upwards in which T(n,k) is the n-th number j such that, descending by the main diagonal of the pyramid described in A245092, the height difference between the level j (starting from the top) and the level of the next terrace is equal to k.

Original entry on oeis.org

1, 3, 2, 5, 4, 9, 7, 6, 12, 20, 8, 10, 21, 36, 72, 11, 13, 25, 50, 91, 144, 14, 16, 32, 56, 112

Offset: 1

Omar E. Pol, Dec 29 2016

This is a permutation of the natural numbers.
Column k lists the numbers with precipice k. For more information about the precipices see A280223 and A280295.
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 m-th level of the pyramid are also the parts of the symmetric representation of sigma(m), m >= 1.
The stepped pyramid is also one of the 3D-quadrants of the stepped pyramid described in A244050.
If a number m is in the column k and k > 1 then m + 1 is the column k - 1.
The largest Dyck path of the symmetric representations of next k - 1 positive integers greater than T(n,k) shares the middle point of the largest Dyck path of the symmetric representation of sigma(T(n,k)). For more information see A237593.

			The corner of the square array begins:
   1,  2,  9, 20, 72, 144,
   3,  4, 12, 36, 91,
   5,  6, 21, 50,
   7, 10, 25,
   8, 13,
  11,
  ...
T(1,6) = 144 because it is the smallest number with precipice 6.

Row 1 gives A280295.
Column 1 gives A276112.
Cf. A000203, A071562, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A240542, A244050, A245092, A259179, A262626, A279286, A279385, A280223.

T(n,1) = A071562(n+1) - 1.

a(20)-a(26) from Omar E. Pol, Jan 02 2017

A280295 Smallest number with precipice n. Descending by the main diagonal of the pyramid described in A245092, the height difference between the level a(n) (starting from the top) and the level of the next terrace is equal to n.

Original entry on oeis.org

1, 2, 9, 20, 72, 144

Offset: 1

Omar E. Pol, Dec 31 2016

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 k-th level of the pyramid are also the parts of the symmetric representation of sigma(k), k >= 1.
The stepped pyramid is also one of the 3D-quadrants of the stepped pyramid described in A244050.
For more information about the precipices see A277437 and A280223.
Is this sequence infinite?

			a(3) = 9 because descending by the main diagonal of the pyramid, the height difference between the level 9 and the level of the next terrace is equal to 3, and 9 is the smallest number with this property.

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

Row 1 of A277437.

Cf. A000203, A196020, A235791, A236104, A237591, A237593, A240542, A244050, A245092, A259179, A262626, A276112, A277437, A279286, A279385, A280223.

a(6) from Omar E. Pol, Jan 02 2017

A276112 Numbers with precipice 1: descending by the main diagonal of the pyramid described in A245092, the height difference between the level a(n) (starting from the top) and the level of the next terrace is equal to 1.

Original entry on oeis.org

1, 3, 5, 7, 8, 11, 14, 15, 17, 19, 23, 24, 27, 29, 31, 34, 35, 39, 41, 44, 47, 48, 49, 53, 55, 59, 62, 63, 65, 69, 71, 76, 79, 80, 83, 87, 89, 90, 95, 97, 98, 99, 103, 107, 109, 111, 116, 119, 120, 125, 127, 129, 131, 134, 139, 142, 143, 149, 152, 153, 155, 159

Offset: 1

Omar E. Pol, Jan 02 2017

nonn

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 k-th level of the pyramid are also the parts of the symmetric representation of sigma(k).
The stepped pyramid is also one of the 3D-quadrants of the stepped pyramid described in A244050.
For more information about the precipices see A277437, A280223 and A280295.
From Hartmut F. W. Hoft, Feb 02 2022: (Start)
Also partial sums of A280919.
a(n) is also the largest number of a Dyck path that crosses the diagonal at point A282131(n) which also is the rightmost number in each nonzero row of the irregular triangle in A279385. (End)

			From _Hartmut F. W. Hoft_, Feb 02 2022: (Start)
      n: 1  2  3  4  5  6  7  8  9 10 11 12 13 14 index.
A282131: 1  2  3  5  6  7  9 11 12 13 15 17 18 20 position on diagonal.
A276112: 1  3  5  7  8 11 14 15 17 19 23 24 27 29 max index of Dyck path.
A280919: 1  2  2  2  1  3  3  1  2  2  4  1  3  2 paths at diag position.
(End)

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

Column 1 of A277437.
Cf. A000203, A071562, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A240542, A244050, A245092, A259179, A262626, A279286, A279385, A280223, A280295.
Cf. A280919, A282131.

(* last computed value of a280919[ ] is dropped to avoid a potential undercount of crossings *)
a240542[n_] := Sum[(-1)^(k+1)Ceiling[(n+1)/k-(k+1)/2], {k, 1, Floor[-1/2+1/2 Sqrt[8n+1]]}]
a280919[n_] := Most[Map[Length, Split[Map[a240542, Range[n]]]]]
A276112[160] (* Hartmut F. W. Hoft, Feb 02 2022 *)

a(n) = A071562(n+1) - 1.

a(n) = Sum_{i=1..n} A280919(i), n >= 1. - Hartmut F. W. Hoft, Feb 02 2022

A299692 a(n) is the total area that is visible in the perspective view of the stepped pyramid with n levels described in A245092.

Original entry on oeis.org

3, 10, 20, 35, 51, 75, 97, 128, 159, 197, 231, 283, 323, 375, 429, 492, 544, 619, 677, 759, 833, 913, 983, 1091, 1172, 1266, 1360, 1472, 1560, 1692, 1786, 1913, 2027, 2149, 2267, 2430, 2542, 2678, 2812, 2982, 3106, 3286, 3416, 3588, 3756, 3920, 4062, 4282, 4437, 4630, 4804, 5006, 5166, 5394, 5576, 5808, 6002

Offset: 1

Omar E. Pol, Mar 06 2018

nonn

a(n) is also the sum of all divisors of all positive integers <= n, plus the n-th oblong number, since A024916(n) equals the total area of the horizontal terraces of the stepped pyramid with n levels, and A002378(n) equals the total area of the vertical sides that are visible (see link).

a(n) is also the sum of all aliquot divisors of all positive integers <= n, plus the n-th triangular matchstick number.

			For n = 3 the areas of the terraces of the first three levels starting from the top of the stepped pyramid are 1, 3 and 4 respectively. On the other hand the areas of the vertical sides that are visible are [1, 1], [2, 2], [2, 1, 1, 2], or in successive levels 2, 4, 6 respectively. Hence the total area that is visible is equal to 1 + 3 + 4 + 2 + 4 + 6 = 8 + 12 = 20, so a(3) = 20.
For n = 16 the total number of horizontal and vertical cells that are visible are 220 and 272 respectively. So a(16) = 220 + 272 = 492 (see the link).

Omar E. Pol, Perspective view of the pyramid with 16 levels, contains 492 cells.

Partial sums of A224880.

Cf. A002378, A024916, A045943, A072691, A153485, A196020, A236104, A237048, A237270, A237271, A237591, A237593, A244050, A245092, A262626, A328366.

Accumulate[Table[DivisorSigma[1, n] + 2*n, {n, 1, 50}]] (* Amiram Eldar, Mar 21 2024 *)

a(n) = sum(k=1, n, n\k*k) + n*(n+1); \\ Michel Marcus, Jun 21 2018

from math import isqrt
def A299692(n): return n*(n+1)+(-(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))>>1) # Chai Wah Wu, Oct 22 2023

a(n) = A024916(n) + A002378(n).
a(n) = A153485(n) + A045943(n).
a(n) = A328366(n)/2. - Omar E. Pol, Apr 22 2020
a(n) = c * n^2 + O(n*log(n)), where c = zeta(2)/2 + 1 = A072691 + 1 = 1.822467... . - Amiram Eldar, Mar 21 2024

A325300 a(n) is the number of faces of the stepped pyramid with n levels described in A245092.

Original entry on oeis.org

6, 9, 15, 20, 24, 31, 35, 42, 49, 59, 63, 72, 76, 84, 95, 106, 110, 121, 125

Offset: 1

Omar E. Pol, Apr 16 2019

To calculate a(n) consider that levels greater than n do not exist.
The shape of the n-th level of the pyramid allows us to know if n is prime (see the Formula section).
For more information about the sequences that we can see in the pyramid see A262626.

			For n = 1 the first level of the stepped pyramid (starting from the top) is a cube, and a cube has six faces, so a(1) = 6.

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

Cf. A325301 (number of edges), A325302 (number of vertices).

Cf. A196020, A215848, A235791, A236104, A237270, A237271, A237591, A237593, A245092, A262626, A299692, A323645, A323648.

a(n) = A325301(n) - A325302(n) + 2 (Euler's formula).
a(n) = A323645(n) + 3.
a(n) = a(n-1) + 4 iff n is a prime > 3 (A215848).

A281012 Zig-zag path that we can find descending by the main diagonal of the pyramid described in A245092.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Feb 11 2017

nonn

The odd-indexed terms are the widths of the successive terraces in the main diagonal of the pyramid.
Conjecture: the odd-indexed terms give A281007, the positive terms in A067742.
The even-indexed terms are the precipices from the successive terraces, descending by the main diagonal of the pyramid.
The even-indexed terms give A280919, the first differences of A071562.
The structure of the pyramid arises after the 90-degree-zig-zag folding of the diagram of the isosceles triangle A237593 (see links).
The mentioned pyramid is also a 3D-quadrant of the pyramid described in A244050.

Omar E. Pol, A pyramid related to the sum-of-divisors function and other integers sequences
Omar E. Pol, Diagram of the isosceles triangle A237593 before the 90-degree-zig-zag folding (rows: 1..28)
Omar E. Pol, Perspective view of the pyramid (first 16 levels)

Also A281007 and A280919 interleaved (conjectured).

Cf. A067742, A071562, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A244050, A245092, A262626, A279387, A280223.

A325301 a(n) is the number of edges of the stepped pyramid with n levels described in A245092.

Original entry on oeis.org

12, 21, 36, 51, 63, 84, 96, 117, 138, 165, 177, 204, 216, 240, 273, 306, 318, 351, 363

Offset: 1

Omar E. Pol, Apr 16 2019

To calculate a(n) consider that levels greater than n do not exist.

			For n = 1 the first level of the stepped pyramid (starting from the top) is a cube, and a cube has 12 edges, so a(1) = 12.

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

Cf. A325300 (number of faces), A325302 (number of vertices).

Cf. A196020, A235791, A236104, A237270, A237271, A237591, A237593, A245092, A262626, A294848, A294849, A317109, A323648.

a(n) = A325300(n) + A325302(n) - 2 (Euler's formula).

A325302 a(n) is the number of vertices of the stepped pyramid with n levels described in A245092.

Original entry on oeis.org

8, 14, 23, 33, 41, 55, 63, 77, 91, 108, 116, 134, 142, 158, 180, 202, 210, 232, 240

Offset: 1

Omar E. Pol, Apr 16 2019

To calculate a(n) consider that levels greater than n do not exist.

			For n = 1 the first level of the stepped pyramid (starting from the top) is a cube, and a cube has 8 vertices, so a(1) = 8.

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

Cf. A325300 (number of faces), A325301 (number of edges).

Cf. A196020, A235791, A236104, A237270, A237271, A237591, A237593, A245092, A262626, A294723, A323648.

a(n) = A325301(n) - A325300(n) + 2 (Euler's formula).

cp's OEIS Frontend

A280919 Precipices from the successive terraces, descending by the main diagonal of the pyramid described in A245092. Also first differences of A071562.

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A280223 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.

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A277437 Square array read by antidiagonals upwards in which T(n,k) is the n-th number j such that, descending by the main diagonal of the pyramid described in A245092, the height difference between the level j (starting from the top) and the level of the next terrace is equal to k.

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A280295 Smallest number with precipice n. Descending by the main diagonal of the pyramid described in A245092, the height difference between the level a(n) (starting from the top) and the level of the next terrace is equal to n.

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A276112 Numbers with precipice 1: descending by the main diagonal of the pyramid described in A245092, the height difference between the level a(n) (starting from the top) and the level of the next terrace is equal to 1.

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A299692 a(n) is the total area that is visible in the perspective view of the stepped pyramid with n levels described in A245092.

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A325300 a(n) is the number of faces of the stepped pyramid with n levels described in A245092.

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A281012 Zig-zag path that we can find descending by the main diagonal of the pyramid described in A245092.

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