A240236 Triangle read by rows: sum of digits of n in base k, for 2<=k<=n.

1, 2, 1, 1, 2, 1, 2, 3, 2, 1, 2, 2, 3, 2, 1, 3, 3, 4, 3, 2, 1, 1, 4, 2, 4, 3, 2, 1, 2, 1, 3, 5, 4, 3, 2, 1, 2, 2, 4, 2, 5, 4, 3, 2, 1, 3, 3, 5, 3, 6, 5, 4, 3, 2, 1, 2, 2, 3, 4, 2, 6, 5, 4, 3, 2, 1, 3, 3, 4, 5, 3, 7, 6, 5, 4, 3, 2, 1, 3, 4, 5, 6, 4, 2, 7, 6, 5, 4, 3, 2, 1

Offset: 2

Author

Franklin T. Adams-Watters, Apr 02 2014

Keywords

• nonn
• tabl
• base
• changed

Examples

			Triangle starts:
  1
  2 1
  1 2 1
  2 3 2 1
  2 2 3 2 1
  3 3 4 3 2 1
		

Links

• Reinhard Zumkeller, Rows n = 2..100 of triangle, flattened

Crossrefs

Row sums give A043306.

Cf. A000120, A053735, A053737, A007953.

See A138530 for another version.

Cf. A002260, A090623.

Programs

Haskell

a240236 n k = a240236_tabl !! (n-1) !! (k-1)
a240236_row n = a240236_tabl !! (n-1)
a240236_tabl = zipWith (map . flip q)
                       [2..] (map tail $ tail a002260_tabl) where
   q b n = if n < b then n else q b n' + d where (n', d) = divMod n b
-- Reinhard Zumkeller, Apr 29 2015

Mathematica

Table[Total[Flatten[IntegerDigits[n,k]]],{n,20},{k,2,n}]//Flatten (* Harvey P. Dale, Jan 13 2025 *)

PARI

T(n,k) = local(r=0);if(k<2,-1,while(n>0,r+=n%k;n\=k);r)

PARI

T(n, k) = sumdigits(n, k) \\ Zhuorui He, Aug 25 2025

Formula

T(n,k) = n - (k - 1) * Sum_{i=1..floor(log_k(n))} floor(n/k^i). - Ridouane Oudra, Sep 27 2024

T(n,k) = n - (k - 1) * A090623(n,k). - Zhuorui He, Aug 25 2025