A276112 -id:A276112 - 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

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

A299472 a(n) is the sum of all divisors of all numbers k whose associated largest Dyck path contains the point (n,n) in the diagram of the symmetric representation of sigma(k) described in A237593, or 0 if no such k exists.

Original entry on oeis.org

1, 7, 13, 0, 20, 15, 43, 0, 66, 0, 24, 49, 59, 0, 134, 0, 60, 113, 0, 86, 0, 104, 165, 0, 48, 245, 0, 132, 0, 224, 0, 198, 0, 124, 57, 317, 0, 192, 0, 350, 0, 326, 0, 104, 211, 0, 434, 0, 216, 0, 0, 647, 0, 344, 0, 186, 331, 0, 584, 0, 270, 0, 234, 0, 672, 0, 350, 171, 0, 156, 639, 0, 672, 0, 390, 0, 368, 0, 956

Offset: 1

Omar E. Pol, Feb 19 2018

nonn

Row sums of A299693.

Cf. A196020, A235791, A236104, A237048, A237591, A237593, A240542, A245092, A259179, A276112, A277437, A279286, A279385, A280919, A280223, A282131, A282197, A280295, A281012.

A299693 Irregular triangle read by rows in which row n lists the total sum of the divisors of all numbers k such that the largest Dyck path of the symmetric representation of sigma(k) contains the point (n,n); or row n is 0 if no such k exists.

Original entry on oeis.org

1, 3, 4, 7, 6, 0, 12, 8, 15, 13, 18, 12, 0, 28, 14, 24, 0, 24, 31, 18, 39, 20, 0, 42, 32, 36, 24, 0, 60, 31, 42, 40, 0, 56, 30, 0, 72, 32, 63, 48, 54, 0, 48, 91, 38, 60, 56, 0, 90, 42, 0, 96, 44, 84, 0, 78, 72, 48, 0, 124, 57, 93, 72, 98, 54, 0, 120, 72, 0, 120, 80, 90, 60, 0, 168, 62, 96, 0, 104, 127, 84, 0

Offset: 1

Omar E. Pol, Feb 19 2018

			Triangle begins:
   1;
   3,  4;
   7,  6;
   0;
  12,  8;
  15;
  13, 18, 12;
   0;
  28, 14, 24;
   0;
  24;
  31, 18;
  39, 20;
   0;
  42, 32, 36, 24;
   0;
...

Nonzero terms give A000203.
Row sums give A299472.
Cf. A259179(n) is the number of positive terms in row n.
Cf. A071562, A196020, A235791, A236104, A237048, A237591, A237593, A240542, A244050, A245092, A276112, A277437, A279286, A279385, A280919, A280223, A282131, A282197, A280295, A281012.

T(n,m) = A000203(A279385(n,m)) if A279385(n,m) > 0, otherwise T(n,m) = 0.

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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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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A299472 a(n) is the sum of all divisors of all numbers k whose associated largest Dyck path contains the point (n,n) in the diagram of the symmetric representation of sigma(k) described in A237593, or 0 if no such k exists.

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A299693 Irregular triangle read by rows in which row n lists the total sum of the divisors of all numbers k such that the largest Dyck path of the symmetric representation of sigma(k) contains the point (n,n); or row n is 0 if no such k exists.

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