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A355212
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A variant of the EKG sequence (A064413) where the least value not yet in the sequence appears as soon as possible.
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2
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1, 2, 6, 3, 12, 4, 10, 5, 35, 7, 14, 8, 18, 9, 33, 11, 143, 13, 39, 15, 20, 16, 34, 17, 323, 19, 57, 21, 24, 22, 46, 23, 115, 25, 30, 26, 36, 27, 42, 28, 58, 29, 899, 31, 62, 32, 74, 37, 148, 38, 40, 82, 41, 1763, 43, 86, 44, 48, 45, 141, 47, 329, 49, 56, 50
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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To build the sequence:
- we start with a(1) = 1 and a(2) = 2, and then repeatedly:
- let a(n) be the last known term and v the least value not yet in the sequence,
- if gcd(a(n), v) > 1
then a(n+1) = v,
- otherwise:
- let w be the least value not yet in the sequence such that gcd(a(n), w) > 1
and gcd(w, v) > 1,
- then a(n+1) = w and a(n+2) = v.
This sequence is a permutation of the positive integers with inverse A355213.
The construction is similar to that of A352713.
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LINKS
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EXAMPLE
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The first terms are (stars correspond to "w" terms):
n a(n) w
-- ---- -
1 1
2 2
3 6 *
4 3
5 12 *
6 4
7 10 *
8 5
9 35 *
10 7
11 14 *
12 8
13 18 *
14 9
15 33 *
16 11
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PROG
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(PARI) See Links section.
(Python)
from math import gcd
from itertools import count, islice
def agen(): # generator of terms
aset, an, v = {1, 2}, 2, 3; yield from [1, 2]
for n in count(3):
if gcd(an, v) == 1:
w = v + 1
while w in aset or gcd(an, w) == 1 or gcd(w, v) == 1: w += 1
aset.add(w); yield w
an = v; aset.add(an); yield an
while v in aset: v += 1
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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