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

Author

N. J. A. Sloane, Dec 30 2021

Keywords

• nonn
• tabf
• base

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.

Examples

			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 = A230624(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.