A010783 -id:A010783 - cp's OEIS frontend

A159861 Square array A(m,n), m>=1, n>=1, read by antidiagonals: A(m,1)=1, A(m,n) is the rank with respect to m of the concatenation of all preceding terms in row m, and the rank of S with respect to m is floor ((S+m-1)/m).

1, 1, 1, 11, 1, 1, 1111, 6, 1, 1, 11111111, 58, 4, 1, 1, 1111111111111111, 5829, 38, 3, 1, 1, 11111111111111111111111111111111, 58292915, 3813, 29, 3, 1, 1, 1111111111111111111111111111111111111111111111111111111111111111, 5829291479146458, 38127938, 2833, 23, 2, 1, 1

Offset: 1

Eric Angelini and Alois P. Heinz, Apr 24 2009

			A(3,4) = 38, because A(3,1).A(3,2).A(3,3) = 114, and the rank of 114 with respect to 3 is floor(116/3) = 38.
Square array A(m,n) begins:
  1,  1, 11, 1111, 11111111, 1111111111111111,  ...
  1,  1,  6,   58,     5829,         58292915,  ...
  1,  1,  4,   38,     3813,         38127938,  ...
  1,  1,  3,   29,     2833,         28323209,  ...
  1,  1,  3,   23,     2265,         22646453,  ...
  1,  1,  2,   19,     1870,         18698645,  ...

Alois P. Heinz, Antidiagonals n = 1..12, flattened

Row m=2 gives: A156147.
Main diagonal gives: A159862.
Cf. A156146, A010783.

R:= (S,m)-> iquo(S+m-1, m):
A:= proc(m, n) option remember; `if`(n=1, 1,
      R(parse(cat(seq(A(m, j), j=1..n-1))), m))
    end:
seq(seq(A(m, d-m), m=1..d-1), d=1..10);

R[S_, m_] := Quotient[S + m - 1, m];
A[m_, n_] := If[n == 1, 1, R[ToExpression@StringJoin[ToString /@ Table[A[m, j], {j, 1, n - 1}]], m]];
Table[Table[A[m, d - m], {m, 1, d - 1}], {d, 1, 10}] // Flatten (* Jean-François Alcover, Feb 13 2023, after Maple code *)

A159862 Main diagonal of A159861.

Original entry on oeis.org

1, 1, 4, 29, 2265, 18698645, 1602308616574727, 14017675267522095175220940844027, 1245902734717669791621141496863001384336371908521990690157218737

Offset: 1

Eric Angelini and Alois P. Heinz, Apr 24 2009

The length (number of decimal digits) of a(n) may be a power of 2 and often simply doubles, when n is increased by 1. But there are many exceptions: n = 11, 12, 13 give lengths 2^8, 3*2^7, 2^9, respectively. A factor of 3 is found in the lengths of a(n) for n = 12, 112..123, 1113..1234, 11123..12345, and so on. A factor of 7 is found for n = 1112, 11112..11122, and so on. 15 is factor of the length of a(11111112).

Alois P. Heinz, Table of n, a(n) for n = 1..12

Cf. A156146, A156147, A000040, A010783.

R:= (S,m)-> iquo(S+m-1, m):
A:= proc(m, n) option remember; `if`(n=1, 1,
      R(parse(cat(seq(A(m, j), j=1..n-1))), m))
    end:
a:= n-> A(n,n):
seq(a(n), n=1..10);

R[S_, m_] := Quotient[S + m - 1, m];
A[m_, n_] := If[n == 1, 1, R[ToExpression@StringJoin[ToString /@ Table[A[m, j], {j, 1, n - 1}]], m]];
a[n_] := A[n, n];
Table[a[n], {n, 1, 10}] (* Jean-François Alcover, Feb 13 2023, after Maple code *)

A004199 Table of [ x/y ], where (x,y) = (1,1),(1,2),(2,1),(1,3),(2,2),(3,1),...

Original entry on oeis.org

1, 0, 2, 0, 1, 3, 0, 0, 1, 4, 0, 0, 1, 2, 5, 0, 0, 0, 1, 2, 6, 0, 0, 0, 1, 1, 3, 7, 0, 0, 0, 0, 1, 2, 3, 8, 0, 0, 0, 0, 1, 1, 2, 4, 9, 0, 0, 0, 0, 0, 1, 1, 2, 4, 10, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 11, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 12, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 3, 6, 13, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 6, 14

Offset: 1

David W. Wilson

Entry in row n and column k is also the number of multiples of k less than or equal to n, n,k >= 1. - L. Edson Jeffery, Aug 31 2014

			Array begins:
  1, 0, 0, 0, 0, 0, 0, 0, ...
  2, 1, 0, 0, 0, 0, 0, 0, ...
  3, 1, 1, 0, 0, 0, 0, 0, ...
  4, 2, 1, 1, 0, 0, 0, 0, ...
  5, 2, 1, 1, 1, 0, 0, 0, ...
  ...

Cf. A002541 (antidiagonal sums).

Cf. A010766 (same sequence as triangle, omitting the zeros), A010783.

(* Array version: *)
Grid[Table[Floor[n/k], {n, 14}, {k, 14}]] (* L. Edson Jeffery, Aug 31 2014 *)
(* Array antidiagonals flattened: *)
Flatten[Table[Floor[(n - k + 1)/k], {n, 14}, {k, n}]] (* L. Edson Jeffery, Aug 31 2014 *)

Sum_{k=1..n} a(n-k+1,k) = A002541(n+1).

cp's OEIS Frontend

A159861 Square array A(m,n), m>=1, n>=1, read by antidiagonals: A(m,1)=1, A(m,n) is the rank with respect to m of the concatenation of all preceding terms in row m, and the rank of S with respect to m is floor ((S+m-1)/m).

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A159862 Main diagonal of A159861.

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A004199 Table of [ x/y ], where (x,y) = (1,1),(1,2),(2,1),(1,3),(2,2),(3,1),...

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