A349822 Irregular triangle T(n,b) (n >= 3, 2 <= b <= A230624(n)/2+1) read by rows. Let m = A230624(n). Then T(n,b) is the smallest nonnegative number k such that k+S_b(k)=m, where S_b(k) is the sum of the digits of k in base b.

7, 7, 8, 7, 5, 11, 11, 10, 9, 12, 10, 7, 19, 17, 20, 17, 16, 17, 18, 15, 20, 16, 11, 35, 34, 31, 33, 29, 31, 33, 31, 28, 29, 30, 25, 32, 26, 34, 27, 36, 28, 19, 58, 58, 55, 57, 56, 55, 52, 51, 49, 51, 53, 49, 57, 52, 46, 47, 48, 49, 50, 41, 52, 42, 54, 43, 56, 44, 58, 45, 60, 46, 31

Offset

3,1

Comments

T(n,b) must be nonzero for all b >= 2, and so this triangle is actually the upper left corner of an array with infinitely long rows (it is believed that there are also infinitely many rows).

Since T(n,b) = m/2 for all b > m/2, we may truncate row n after m/2 terms. The rows do not change beyond that point.

Links

Table of n, a(n) for n=3..75.

Example

Triangle begins as follows:
   n    m   Row n
   3   10   [7, 7, 8, 7, 5],
   4   14   [11, 11, 10, 9, 12, 10, 7],
   5   22   [19, 17, 20, 17, 16, 17, 18, 15, 20, 16, 11],
   6   38   [35, 34, 31, 33, 29, 31, 33, 31, 28, 29, 30, 25, 32, 26, 34, 27, 36, 28, 19],
   7   62   [58, 58, 55, 57, 56, 55, 52, 51, 49, 51, 53, 49, 57, 52, 46, 47, 48, 49, 50, 41, 52, 42, 54, 43, 56, 44, 58, 45, 60, 46, 31],
   8   94   [90, 89, 89, 87, 87, 83, 89, 79, 83, 82, 80, 77, 86, 82, 77, 79, 81, 74, 85, 77, 89, 80, 70, 71, 72, 73, 74, 75, 76, 77, 78, 63, 80, 64, 82, 65, 84, 66, 86, 67, 88, 68, 90, 69, 92, 70, 47],
...
For n = 3, m = A230264(3) = 10, and row 3 of the triangle is [7, 7, 8, 7, 5], corresponding to the identities (where x_b is the base-b expansion of x):
   10 = 111_2 + 3 = 7 + 3,
      = 21_3 + 3 = 7 + 3
      = 20_4 + 2 = 8 + 2
      = 12_5 + 3 = 7 + 3
      = 5_b + 5  = 5 + 5 for all b >= 6.

Crossrefs

Cf. A230624.

Sequence in context: A198992 A201668 A021853 * A092616 A260634 A294968

Adjacent sequences: A349819 A349820 A349821 * A349823 A349824 A349825

Keyword

nonn,tabf,base

Author

N. J. A. Sloane, Dec 30 2021

Status

approved