A282197 -id:A282197 - cp's OEIS frontend

A279286 a(n) = d if the point (d,d) is shared by a record of different Dyck paths in the main diagonal of the diagram of the symmetries of sigma described in A237593.

1, 2, 7, 15, 52, 102, 296, 371, 455, 929, 1853, 2034, 4517, 4797, 5829, 6146, 6948, 17577, 18915, 60349, 78369, 85171, 123788, 128596, 415355, 906771, 1308771, 3329668

Offset: 1

Omar E. Pol, Dec 09 2016

Is this sequence infinite?
First differs from A282197 (another version) at a(19). - Omar E. Pol, Feb 08 2017
a(n) = d if the point (d,d) belongs to a vertical-line-segment whose length is a record in the main diagonal of the pyramid described in A245092 (starting from the top). The diagram of the symmetries of sigma is also the top view of the mentioned pyramid. See examples. - Omar E. Pol, Feb 09 2017

			The first record of height difference is between the levels 1 and 2 of the pyramid (starting from the top), at the point (1,1) of the main diagonal of the top view of the pyramid, so a(1) = 1.
The second record of height difference is between the levels 2 and 4, at the point (2,2) of the main diagonal of the top view of the pyramid, so a(2) = 2.
The third record of height difference is between the levels 9 and 12, at the point (7,7) of the main diagonal of the top view of the pyramid, so a(3) = 7.
The fourth record of height difference is between the levels 20 and 24, at the point (15,15) of the main diagonal of the top view of the pyramid, so a(4) = 15.
Illustration of the diagram of the symmetries of sigma (n = 1..16), which is also the top view of the pyramid described in A245092, and it is also a quadrant of the top view of the pyramid described in A244050:
.     _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.    |_| | | | | | | | | | | | | | | |
.    |_ _|_| | | | | | | | | | | | | |
.    |_ _|  _|_| | | | | | | | | | | |
.    |_ _ _|    _|_| | | | | | | | | |
.    |_ _ _|  _|  _ _|_| | | | | | | |
.    |_ _ _ _|  _| |  _ _|_| | | | | |
.    |_ _ _ _| |_ _|_|    _ _|_| | | |
.    |_ _ _ _ _|  _|     |  _ _ _|_| |
.    |_ _ _ _ _| |      _|_| |  _ _ _|
.    |_ _ _ _ _ _|  _ _|    _| |
.    |_ _ _ _ _ _| |  _|  _|  _|
.    |_ _ _ _ _ _ _| |_ _|  _|
.    |_ _ _ _ _ _ _| |  _ _|
.    |_ _ _ _ _ _ _ _| |
.    |_ _ _ _ _ _ _ _| |
.    |_ _ _ _ _ _ _ _ _|
...

Where records occur in A259179, (was the original Name).

Cf. A196020, A235791, A236104, A237048, A237270, A237591, A237593, A240542, A245092, A244050, A262626, A282197.

a240542[n_] := Sum[(-1)^(k+1)*Ceiling[(n+1)/k - (k+1)/2], {k, 1, Floor[(Sqrt[8n+1]-1)/2]}]
a279286[b_] := Module[{centers={{1, 1}}, acc={1}, k=2, cPrev=1, cCur, len}, While[k<=b, cCur=a240542[k]; If[Last[acc]==cCur, AppendTo[acc, cCur], len=Length[acc]; If[First[Last[centers]]Hartmut F. W. Hoft, Feb 08 2017 *)

a(7)-a(28) from Hartmut F. W. Hoft, Feb 08 2017

New Name from Omar E. Pol, Feb 09 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

A279286 a(n) = d if the point (d,d) is shared by a record of different Dyck paths in the main diagonal of the diagram of the symmetries of sigma described in A237593.

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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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