A276005 Numbers with hit-free factorial base representations; positions of zeros in A276004 & A276007.

0, 1, 2, 4, 5, 6, 7, 12, 14, 16, 18, 19, 20, 22, 23, 24, 25, 26, 28, 29, 48, 49, 54, 55, 60, 66, 67, 72, 74, 76, 78, 84, 86, 88, 90, 92, 94, 96, 97, 98, 100, 101, 102, 103, 108, 110, 112, 114, 115, 116, 118, 119, 120, 121, 122, 124, 125, 126, 127, 132, 134, 136, 138, 139, 140, 142, 143, 240, 241, 242, 244, 245, 264, 265, 266, 268, 269, 288, 289, 312, 314, 316

Offset

0,3

Comments

We say there is a "hit" in factorial base representation (A007623) of n when there is any such pair of nonzero digits d_i and d_j in positions i > j so that (i - d_i) = j. Here the rightmost (least significant digit) occurs at position 1. This sequence gives all "hit-free" numbers, meaning that for every nonzero digit d_i (in position i) in their factorial base representation the digit at the position (i - d_i) is 0.

Also numbers n for which A060502(n) = A060128(n), in other words, the numbers n for which the number of slopes in their factorial base representation (A007623) is equal to the number of non-singleton cycles of the permutation listed as n-th permutation in the list A060117 (or A060118).

This can be viewed as a factorial base analog of base-2 related A003714.

Links

Antti Karttunen, Table of n, a(n) for n = 0..4140

Index entries for sequences related to factorial base representation

Formula

Other identities. For all n >= 1:

a(A000110(n)) = n! = A000142(n). [To be proved.]

Example

n=14 (factorial base "210") is included because 2 occurs in position 3 and 1 occurs in position 2, thus as (3-2) = 1 <> 2, 2 does not "hit" digit 1.
n=15 ("211") is NOT included because 2 occurring in position 3 hits the rightmost 1 in position 1 (as 3-2 = 1), and moreover, also the middle 1 hits the rightmost 1 as 2-1 = 1.

Prog

(Scheme, with Antti Karttunen's IntSeq-library)
(define A276005 (ZERO-POS 0 0 A276004))

Crossrefs

Complement: A276006.

Cf. A000110, A000142, A060128, A060502, A276004, A276007.

Cf. A060112 (a subsequence).

Intersection with A275804 gives A261220.

Cf. also A003714, A060117 and A060118.

Sequence in context: A300861 A039057 A317185 * A092058 A134532 A282278

Adjacent sequences: A276002 A276003 A276004 * A276006 A276007 A276008

Keyword

nonn,base

Author

Antti Karttunen, Aug 17 2016

Status

approved