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

1,2

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).

Links

Amiram Eldar, Table of n, a(n) for n = 1..1000

Wikipedia, Factorial number system.

Example

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.

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]

Crossrefs

A065355 is a subsequence.

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

Sequence in context: A189079 A045218 A177856 * A343477 A296220 A297835

Adjacent sequences: A343473 A343474 A343475 * A343477 A343478 A343479

Keyword

nonn,base

Author

Amiram Eldar, Apr 16 2021

Status

approved