A231716 Numbers with restricted residue set factorial base representation: numbers n which can be formed as a sum n = du*u! + ... + d2*2! + d1*1!, where each di is in range 1..i and gcd(di,i+1)=1.

1, 3, 5, 9, 11, 21, 23, 33, 35, 45, 47, 57, 59, 69, 71, 81, 83, 93, 95, 105, 107, 117, 119, 153, 155, 165, 167, 177, 179, 189, 191, 201, 203, 213, 215, 225, 227, 237, 239, 633, 635, 645, 647, 657, 659, 669, 671, 681, 683, 693, 695, 705, 707, 717, 719, 873, 875

Offset

1,2

Comments

A001088(n+1) gives the number of terms x in sequence for which A084558(x)=n.

Because totatives (the reduced residue set) of each natural number k form a multiplicative group of integers modulo same k, it means that taking e.g. inverses of each digit modulo same k or multiplying them (again modulo k) by some member of that set keeps the set closed, and thus applying these operations to each digit modulo i+1 (2 for the least significant digit in factorial base, 3 for the next, and so on) yield only digits allowed in this sequence, and thus they induce various permutations of this sequence. These can be further "normalized" to be permutations of natural numbers with a suitable ranking function, which is to be submitted later.

Links

Antti Karttunen, Rows 1..7, flattened

Wikipedia, Totative

Example

This can be viewed as an irregular table, where row n has A001088(n+1) elements, starts from position A231721(n) and ends at position A231722(n+1):
1;
3, 5;
9, 11, 21, 23;
33, 35, 45, 47, 57, 59, 69, 71, 81, 83, 93, 95, 105, 107, 117, 119;
...

Prog

(Scheme, with Antti Karttunen's IntSeq-library)
(define A231716 (MATCHING-POS 1 1 (lambda (k) (= 1 (A231715 k)))))

Crossrefs

Positions of ones in A231715.

The first term of each row: A007489(n) = a(A231721(n)).

The last term of each row: A033312(n+1) = a(A231722(n+1)).

Subset of A227157.

Cf. also A007623, A000010, A001088, A084558.

Sequence in context: A364808 A187753 A339638 * A113488 A092917 A256220

Adjacent sequences: A231713 A231714 A231715 * A231717 A231718 A231719

Keyword

nonn,tabf

Author

Antti Karttunen, Nov 12 2013

Status

approved