%I #18 Nov 29 2013 22:20:04
%S 1,3,5,9,11,21,23,33,35,45,47,57,59,69,71,81,83,93,95,105,107,117,119,
%T 153,155,165,167,177,179,189,191,201,203,213,215,225,227,237,239,633,
%U 635,645,647,657,659,669,671,681,683,693,695,705,707,717,719,873,875
%N 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.
%C A001088(n+1) gives the number of terms x in sequence for which A084558(x)=n.
%C 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.
%H Antti Karttunen, <a href="/A231716/b231716.txt">Rows 1..7, flattened</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Totative">Totative</a>
%e 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):
%e 1;
%e 3, 5;
%e 9, 11, 21, 23;
%e 33, 35, 45, 47, 57, 59, 69, 71, 81, 83, 93, 95, 105, 107, 117, 119;
%e ...
%o (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o (define A231716 (MATCHING-POS 1 1 (lambda (k) (= 1 (A231715 k)))))
%Y Positions of ones in A231715.
%Y The first term of each row: A007489(n) = a(A231721(n)).
%Y The last term of each row: A033312(n+1) = a(A231722(n+1)).
%Y Subset of A227157.
%Y Cf. also A007623, A000010, A001088, A084558.
%K nonn,tabf
%O 1,2
%A _Antti Karttunen_, Nov 12 2013