A347981 Irregular triangle T(n, k) read by rows in which row n lists the number of parts in the symmetric representation of sigma for n = 2^m * q, 2^(m-1) * q, ... , q, with m >= 0, q odd, 1 <= k <= m + 1.

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

Offset

1,4

Comments

The length of row n = 2^m * q, m>=0, q odd, in the triangle is m+1, i.e., the exponent of the even part of n plus 1. The rightmost number in row n gives the number of regions in the symmetric representation of sigma, for short "#rsrs", of the odd part q of n, its index in this sequence is A005187(n).

The numbers in each row of the triangle are nondecreasing.

The lengths of at least the first 21 rows are given by A001511. - Omar E. Pol, Sep 22 2021

It appears that row lengths give A001511, the columns are A237271, and row 2^i, i >= 0, lists 1 + i ones. Omar E. Pol, Oct 01 2021

Links

Table of n, a(n) for n=1..97.

Formula

T(n, k) = A237271(n/2^(k-1)) for n=2^m*q, m>=0, q odd, 1<=k<=m+1, and

T(n, m+1) = a(A005187(n)) = a(A011371(n) + n), n >= 1.

Example

Row 15, a(26) = (3), consists of a single number since 15 is odd.
Row 48, a(90..94) = (1, 1, 1, 1, 2), is the sequence of #rsrs for the numbers 48, 24, 12, 6, 3.
Row 228, a(450..452) = (1, 2, 4), is the sequence of #rsrs for the numbers 228, 114, 57.
First 21 rows of the triangle; columns indicate division of n by powers of 2:
  n       1  2  4  8  16
-------------------------------------
  1:      1;
  2:      1, 1;
  3:      2;
  4:      1, 1, 1;
  5:      2;
  6:      1, 2;
  7:      2;
  8:      1, 1, 1, 1;
  9:      3;
  10:     2, 2;
  11:     2;
  12:     1, 1, 2;
  13:     2;
  14:     2, 2;
  15:     3;
  16:     1, 1, 1, 1, 1;
  17:     2;
  18:     1, 3;
  19:     2;
  20:     1, 2, 2;
  21:     4;

Mathematica

(* function a237270[ ] is defined in A237270 *)
a237271[n_] := Length[a237270[n]]
a347981[n_] := Last[Transpose[NestWhileList[{First[#]/2, a237271[First[#]/2]}&, {n, a237271[n]}, IntegerExponent[First[#], 2]>0&]]]
Flatten[Map[a347981, Range[50]]]

Crossrefs

Column 1 gives A237271.

Cf. A001511, A005187, A011371, A235791, A237270, A237591, A237593, A279387.

Sequence in context: A072085 A054868 A352517 * A065081 A366643 A212185

Adjacent sequences: A347978 A347979 A347980 * A347982 A347983 A347984

Keyword

nonn,tabf

Author

Hartmut F. W. Hoft, Sep 22 2021

Status

approved