

A330779


Lexicographically earliest sequence of positive integers such that for any v > 0, the value v appears up to v times, and the associate function f defined by f(n) = Sum_{k = 1..n} a(k) * i^k for n >= 0 is injective (where i denotes the imaginary unit).


4



1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 4, 6, 5, 5, 6, 5, 5, 7, 6, 6, 7, 6, 6, 8, 7, 7, 7, 8, 7, 7, 8, 8, 8, 10, 9, 8, 8, 9, 8, 10, 10, 9, 11, 9, 9, 11, 9, 9, 10, 9, 9, 12, 10, 11, 11, 11, 12, 10, 13, 10, 10, 13, 10, 10, 12, 11, 11, 12, 11, 11, 11, 14, 11, 13, 12, 13
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OFFSET

1,2


COMMENTS

The variant of this sequence where each value can only appear up to once, twice or three times corresponds to A000027, A008619 and A008620 respectively.
Graphically, the representation of f resembles a windmill; the variant of f where we allow the value v to appear 3*v times resembles a butterfly (see illustrations in Links section).


LINKS

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of first steps
Rémy Sigrist, Colored representation of f(n) for n = 0..1000000 in the complex plane (where the color is function of n)
Rémy Sigrist, Colored representation of the variant where the value v can appear up to 3*v times
Rémy Sigrist, Colored representation of the variant where the value v can appear up to A000265(v) times
Rémy Sigrist, Colored representation of the variant where the value v can appear up to prime(v) times
Rémy Sigrist, PARI program for A330779


EXAMPLE

The first terms, alongside the corresponding values of f(n), are:
n a(n) f(n)
  
0 N/A 0
1 1 i
2 2 2+i
3 2 2i
4 3 1i
5 3 1+2*i
6 3 2+2*i
7 4 22*i
8 4 22*i
9 4 2+2*i
10 5 3+2*i
11 4 32*i
12 6 32*i
See also illustration in Links section.


PROG

(PARI) See Links section.


CROSSREFS

See A331002 and A331003 for the real and imaginary parts of f, respectively.
See A330780 for another variant.
Cf. A000265.
Sequence in context: A091497 A005707 A087828 * A110867 A006670 A132914
Adjacent sequences: A330776 A330777 A330778 * A330780 A330781 A330782


KEYWORD

nonn


AUTHOR

Rémy Sigrist, Dec 31 2019


STATUS

approved



