A343476 Numbers k whose representations in factorial base include each of the digits from 0 to d-1 exactly once, where d = A084558(k) is the number of digits of k in factorial base.

0, 2, 10, 13, 14, 46, 67, 68, 77, 82, 85, 86, 238, 355, 356, 461, 466, 469, 470, 503, 526, 547, 548, 557, 562, 565, 566, 1438, 2155, 2156, 2861, 2866, 2869, 2870, 3503, 3526, 3547, 3548, 3557, 3562, 3565, 3566, 3719, 3838, 3955, 3956, 4061, 4066, 4069, 4070, 4103

Offset: 1

Author

Amiram Eldar, Apr 16 2021

Keywords

• nonn
• base

Comments

The number of terms with k > 1 digits in factorial base is 2^(k-1) - 1 = A000225(k-1).

The number of terms below k!, for k >= 1, is 2^(k-1) - (k-1) = A000325(k-1).

Examples

			2 is a term since its factorial base representation is {1, 0}.
10, 13 and 14 are terms since their factorial base representations are {1, 2, 0}, {2, 0, 1} and {2, 1, 0}, respectively.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000
• Wikipedia, Factorial number system.

Crossrefs

A065355 is a subsequence.

Cf. A000225, A000325, A007623, A084558, A343477.

Programs

Mathematica

m = 7; bases = Reverse @ Range[2, m]; max = Times @@ bases; factBase[n_] := IntegerDigits[n, MixedRadix[bases]]; q[n_] := Union[(fd = factBase[n])] == Range[0, Length[fd] - 1]; Select[Range[0, max], q]