A301652 Triangle read by rows: row n gives the digits of n in factorial base in reversed order.

0, 1, 0, 1, 1, 1, 0, 2, 1, 2, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 2, 1, 1, 2, 1, 0, 0, 2, 1, 0, 2, 0, 1, 2, 1, 1, 2, 0, 2, 2, 1, 2, 2, 0, 0, 3, 1, 0, 3, 0, 1, 3, 1, 1, 3, 0, 2, 3, 1, 2, 3, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 2, 0, 1, 1, 2, 0, 1, 0, 0, 1, 1

Offset: 0

Author

Seiichi Manyama, Mar 25 2018

Keywords

• nonn
• tabf
• base

Comments

Row n gives exponents for successive primes 2, 3, 5, 7, 11, etc., in the prime factorization of A276076(n). - Antti Karttunen, Mar 11 2024

Examples

			   n | 1  2  6
  ---+---------
   0 | 0;
   1 | 1;
   2 | 0, 1;
   3 | 1, 1;
   4 | 0, 2;
   5 | 1, 2;
   6 | 0, 0, 1;
   7 | 1, 0, 1;
   8 | 0, 1, 1;
   9 | 1, 1, 1;
  10 | 0, 2, 1;
  11 | 1, 2, 1;
  12 | 0, 0, 2;
  13 | 1, 0, 2;
  14 | 0, 1, 2;
  15 | 1, 1, 2;
  16 | 0, 2, 2;
  17 | 1, 2, 2;
  18 | 0, 0, 3;
  19 | 1, 0, 3;
		

Links

• Seiichi Manyama, Rows n = 0..2000, flattened
• Wikipedia, Factorial number system.
• Index entries for sequences related to factorial base representation.

Crossrefs

Triangle A108731 with rows reversed.

Cf. A007623, A034968 (row sums), A208575 (row products), A227153 (products of nonzero terms on row n), A276076, A301593.

Programs

Mathematica

row[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, AppendTo[s, r]; m++]; s]; row[0] = {0}; Array[row, 31, 0] // Flatten (* Amiram Eldar, Mar 11 2024 *)

Sage

terms=25; print([0]+[x for sublist in [[floor(n/factorial(i))%(i+1) for i in [k for k in [1..n] if factorial(k)<=n]] for n in [1..terms]] for x in sublist]) # Tom Edgar, Aug 15 2018

Formula

T(n,k) = floor(n/k!) mod k+1. - Tom Edgar, Aug 15 2018