A381965 -id:A381965 - cp's OEIS frontend

A381966 Row sums of A381965.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 20, 23, 26, 29, 32, 35, 40, 45, 50, 55, 30, 34, 38, 42, 48, 55, 62, 60, 70, 84, 40, 45, 50, 57, 66, 65, 78, 97, 86, 111, 50, 56, 62, 73, 74, 90, 86, 112, 98, 124, 60, 67, 76, 89, 96, 95, 128, 117

Offset: 0

Paolo Xausa, Mar 12 2025

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A031346, A381965.

A381966[n_] := Total[NestWhileList[Times @@ IntegerDigits[#] &, n, # >= 10 &]];
Array[A381966, 100, 0]

a(n) = Sum_{k = 0..A031346(n)} A381965(n,k).

A381962 Irregular triangle read by rows, where row n lists the iterates of f(x), starting at x = n until f(x) <= 1, where f(x) is the Hamming weight of x (A000120).

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 1, 4, 1, 5, 2, 1, 6, 2, 1, 7, 3, 2, 1, 8, 1, 9, 2, 1, 10, 2, 1, 11, 3, 2, 1, 12, 2, 1, 13, 3, 2, 1, 14, 3, 2, 1, 15, 4, 1, 16, 1, 17, 2, 1, 18, 2, 1, 19, 3, 2, 1, 20, 2, 1, 21, 3, 2, 1, 22, 3, 2, 1, 23, 4, 1, 24, 2, 1, 25, 3, 2, 1, 26, 3, 2, 1

Offset: 0

Paolo Xausa, Mar 11 2025

			Triangle begins:
  n\k|  0  1  2  3
  ----------------
   0 |  0;
   1 |  1;
   2 |  2, 1;
   3 |  3, 2, 1;
   4 |  4, 1;
   5 |  5, 2, 1;
   6 |  6, 2, 1;
   7 |  7, 3, 2, 1;
   8 |  8, 1;
   9 |  9, 2, 1;
  10 | 10, 2, 1;
  ...

Paolo Xausa, Table of n, a(n) for n = 0..11724 (rows 0..3000 of triangle, flattened).

Cf. A000120, A078627 (row lengths), A078677 (row sums), A180094.

Cf. A381963, A381965.

A381962row[n_] := NestWhileList[DigitSum[#, 2] &, n, # > 1 &];
Array[A381962row, 30, 0]

def row(n):
    out = [n] if n > 1 else []
    while (n:=n.bit_count()) > 1:
        out += [n]
    return out + [n]
print([e for n in range(27) for e in row(n)]) # Michael S. Branicky, Mar 12 2025

T(n,0) = n and, for k = 1..A180094(n), T(n,k) = A000120(T(n,k-1)).

A381963 Irregular triangle read by rows, where row n lists the iterates of f(x), starting at x = n until f(x) < 10, where f(x) is the digital sum of x (A007953).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 11, 2, 12, 3, 13, 4, 14, 5, 15, 6, 16, 7, 17, 8, 18, 9, 19, 10, 1, 20, 2, 21, 3, 22, 4, 23, 5, 24, 6, 25, 7, 26, 8, 27, 9, 28, 10, 1, 29, 11, 2, 30, 3, 31, 4, 32, 5, 33, 6, 34, 7, 35, 8, 36, 9, 37, 10, 1, 38, 11, 2, 39, 12, 3

Offset: 0

Paolo Xausa, Mar 11 2025

			Triangle begins:
  n\k|  0   1   2
  ---------------
   0 |  0;
   1 |  1;
   2 |  2;
   3 |  3;
   4 |  4;
   5 |  5;
   6 |  6;
   7 |  7;
   8 |  8;
   9 |  9;
  10 | 10,  1;
  11 | 11,  2;
  12 | 12,  3;
  13 | 13,  4;
  14 | 14,  5;
  15 | 15,  6;
  16 | 16,  7;
  17 | 17,  8;
  18 | 18,  9;
  19 | 19, 10,  1;
  20 | 20,  2;
  ...

Paolo Xausa, Table of n, a(n) for n = 0..11648 (rows 0..4000 of triangle, flattened).

Cf. A007953, A010888 (right border), A031286 (row lengths - 1), A381964 (row sums).

Cf. A381962, A381965.

A381963row[n_] := NestWhileList[DigitSum, n, # >= 10 &];
Array[A381963row, 40, 0]

T(n,0) = n and, for k = 1..A031286(n), T(n,k) = A007953(T(n,k-1)).

cp's OEIS Frontend

A381966 Row sums of A381965.

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A381962 Irregular triangle read by rows, where row n lists the iterates of f(x), starting at x = n until f(x) <= 1, where f(x) is the Hamming weight of x (A000120).

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A381963 Irregular triangle read by rows, where row n lists the iterates of f(x), starting at x = n until f(x) < 10, where f(x) is the digital sum of x (A007953).

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Crossrefs

Programs

Mathematica

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