A231716 - OEIS

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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

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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):

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)))))