A138530 Triangle read by rows: T(n,k) = sum of digits of n in base k representation, 1<=k<=n.

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

Offset: 1

Author

Reinhard Zumkeller, Mar 26 2008

Keywords

• nonn
• base
• tabl

Comments

A131383(n) = sum of n-th row;

A000027(n) = T(n,1);

A000120(n) = T(n,2) for n>1;

A053735(n) = T(n,3) for n>2;

A053737(n) = T(n,4) for n>3;

A053824(n) = T(n,5) for n>4;

A053827(n) = T(n,6) for n>5;

A053828(n) = T(n,7) for n>6;

A053829(n) = T(n,8) for n>7;

A053830(n) = T(n,9) for n>8;

A007953(n) = T(n,10) for n>9;

A053831(n) = T(n,11) for n>10;

A053832(n) = T(n,12) for n>11;

A053833(n) = T(n,13) for n>12;

A053834(n) = T(n,14) for n>13;

A053835(n) = T(n,15) for n>14;

A053836(n) = T(n,16) for n>15;

A007395(n) = T(n,n-1) for n>1;

A000012(n) = T(n,n).

Examples

			Start of the triangle for n in base k representation:
......................1
....................11....10
......... ........111....11...10
................1111...100...11..10
..............11111...101...12..11..10
............111111...110...20..12..11..10
..........1111111...111...21..13..12..11..10
........11111111..1000...22..20..13..12..11..10
......111111111..1001..100..21..14..13..12..11..10
....1111111111..1010..101..22..20..14..13..12..11..10
..11111111111..1011..102..23..21..15..14..13..12..11..10
111111111111..1100..110..30..22..20..15..14..13..12..11..10,
and the triangle of sums of digits starts:
......................1
.....................2...1
......... ..........3...2...1
...................4...1...2...1
..................5...2...3...2...1
.................6...2...2...3...2...1
................7...3...3...4...3...2...1
...............8...1...4...2...4...3...2...1
..............9...2...1...3...5...4...3...2...1
............10...2...2...4...2...5...4...3...2...1
...........11...3...3...5...3...6...5...4...3...2...1
..........12...2...2...3...4...2...6...5...4...3...2...1.
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• Eric Weisstein's World of Mathematics, Digit Sum

Crossrefs

Cf. A007953. See A240236 for another version.

Cf. A002260.

Programs

Haskell

a138530 n k = a138530_tabl !! (n-1) !! (k-1)
a138530_row n = a138530_tabl !! (n-1)
a138530_tabl = zipWith (map . flip q) [1..] a002260_tabl where
   q 1 n = n
   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

T[n_, k_] := If[k == 1, n, Total[IntegerDigits[n, k]]];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 25 2021 *)