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A101387 Quet transform of A002260. 2
1, 1, 2, 1, 3, 1, 5, 1, 1, 7, 1, 2, 1, 9, 1, 3, 1, 12, 1, 1, 4, 1, 15, 1, 2, 1, 5, 1, 18, 1, 3, 1, 7, 1, 1, 21, 1, 4, 1, 9, 1, 2, 1, 24, 1, 5, 1, 11, 1, 3, 1, 28, 1, 1, 6, 1, 13, 1, 4, 1, 32, 1, 2, 1, 7, 1, 15, 1, 5, 1, 36, 1, 3, 1, 9, 1, 1, 17, 1, 6, 1, 40, 1, 4, 1, 11, 1, 2, 1, 19, 1, 7, 1, 44, 1, 5, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The Quet transform converts any sequence of positive integers containing an infinite number of 1's into another sequence of positive integers containing an infinite number of 1's.

Start with a sequence, {a(k)}, of only positive integers and an infinite number of 1's. Example: 1,1,2,1,2,3,1,2,3,4,1,... (A002260).

Form the sequence {b(k)} (which is a permutation of the positive integers), given by b(k) = the a(k)th smallest positive integer not yet in the sequence b, with b(1)=a(1).

In the example b is 1,2,4,3,6,8,5,9,11,13,7,12,15,... (A065562).

Let {c(k)} be the inverse of {b(k)}. In the example c = 1,2,4,3,7,5,11,6,8,16,9,12... (A065579).

Form the final sequence {d(k)}, where each d(k) is such that c(k) = the d(k)th smallest positive integer not yet in the sequence c, with d(1)=c(1).

In the example d is 1,1,2,1,3,1,5,1,1,7,1,2,1,9,1,3,1,12,1,1,4,1,15,... (the current sequence).

A more formal description of the Quet transform is as follows.

Let N denote the positive integers. For any permutation p: N -> N, let T(p): N -> N be given by T(p)(n) = # of elements in {m in N | m >= n AND p(m) <= p(n)}. Observe that T is a bijection from the set of permutations N -> N onto the set of sequences N -> N that contain infinitely many 1's.

Now suppose f: N -> N contains infinitely many 1's; then its Quet transform Q(f): N -> N is T^(-1)[(T(f))^(-1)], which also contains infinitely many 1's. Q is self-inverse; f and Q(f) correspond via T to a permutation and its inverse.

LINKS

Table of n, a(n) for n=1..97.

David Wasserman, Quet Transform

CROSSREFS

Cf. A002260, A100661.

Sequence in context: A110977 A069230 A163961 * A117365 A116212 A079068

Adjacent sequences:  A101384 A101385 A101386 * A101388 A101389 A101390

KEYWORD

easy,nonn,less

AUTHOR

David Wasserman, Jan 14 2005

STATUS

approved

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Last modified April 16 02:33 EDT 2014. Contains 240534 sequences.