A301652 - OEIS

%I #52 Mar 11 2024 08:31:31

%S 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,

%T 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,

%U 1,0,0,1,0,1,0,1,1,1,0,1,0,2,0,1,1,2,0,1,0,0,1,1

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

%C 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

%H Seiichi Manyama, <a href="/A301652/b301652.txt">Rows n = 0..2000, flattened</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Factoradic">Factorial number system</a>.

%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>.

%F T(n,k) = floor(n/k!) mod k+1. - _Tom Edgar_, Aug 15 2018

%e    n | 1  2  6

%e   ---+---------

%e    0 | 0;

%e    1 | 1;

%e    2 | 0, 1;

%e    3 | 1, 1;

%e    4 | 0, 2;

%e    5 | 1, 2;

%e    6 | 0, 0, 1;

%e    7 | 1, 0, 1;

%e    8 | 0, 1, 1;

%e    9 | 1, 1, 1;

%e   10 | 0, 2, 1;

%e   11 | 1, 2, 1;

%e   12 | 0, 0, 2;

%e   13 | 1, 0, 2;

%e   14 | 0, 1, 2;

%e   15 | 1, 1, 2;

%e   16 | 0, 2, 2;

%e   17 | 1, 2, 2;

%e   18 | 0, 0, 3;

%e   19 | 1, 0, 3;

%t 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 *)

%o (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

%Y Triangle A108731 with rows reversed.

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

%K nonn,tabf,base

%O 0,8

%A _Seiichi Manyama_, Mar 25 2018