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A374349
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Integers >=0 whose decimal digits are topologically distinct from those of any smaller number.
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0
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0, 1, 8, 10, 11, 18, 40, 48, 88, 100, 101, 108, 111, 118, 188, 400, 408, 488, 888, 1000, 1001, 1008, 1011, 1018, 1088, 1111, 1118, 1188, 1888, 4000, 4008, 4088, 4888, 8888, 10000, 10001, 10008, 10011, 10018, 10088, 10111, 10118, 10188, 10888, 11111, 11118
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OFFSET
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1,3
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COMMENTS
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Assumes 0 without a slash or a center dot, closed 4, 6, and 9, and no overlapping of multiple digits. Digits homologous to a (flattened) sphere: 1, 2, 3, 5, 7; to a torus: 0, 4, 6, 9; to a double torus: 8. Sequence is a run of the terms in ascending numeric order.
All topologically distinct terms can be represented by nondecreasing sequences of strings of 0s, 1s, and 8s. However, terms cannot begin with 0. Therefore, if a string has 0s, then (i) if there are any 1s, one of them moves to the front, (ii) else, the first 0 is replaced with 4. Sequence is the resulting strings sorted as base-10 numbers. - Michael S. Branicky, Jul 11 2024
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LINKS
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EXAMPLE
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0 is homologous to 1 torus, so a(1)=0.
1 is homologous to 1 sphere, so a(2)=1.
2 is homologous to 1 sphere, same as 1, so it is not in the sequence.
4 is homologous to 1 torus, same as 0, so it is not in the sequence.
8 is homologous to 1 double torus, so a(3)=8.
10 is homologous to 1 sphere and 1 torus, so a(4)=10.
11 is homologous to 2 spheres, so a(5)=11.
14 is homologous to 1 sphere and 1 torus, same as 10, so it is not in the sequence.
41 is homologous to 1 sphere and 1 torus, same as 10, so it is not in the sequence.
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PROG
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(PARI) df(d, c)=(10^c-1)/9*d
n=0; a=0; at=1; while(true, a++; at+=a+1; ac=0; for(b=0, a, for(c=0, b, n++; print(n, " ", if(n<=2, n-1, ac+b-c+1<at, 10^(a-1)+df(1, b)+df(7, c), ac+1<at, 4*10^(a-1)+df(8, c), df(8, a))); if(n==10000, break(3)); ac++)))
(Python)
from itertools import count, islice, combinations_with_replacement as cwr
def agen(): # generator of terms
after = {"1":"018", "4":"08", "8":"8"}
yield from (0, 1, 8)
for digits in count(2):
for first in "148":
for rest in cwr(after[first], digits-1):
yield int(first + "".join(rest))
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CROSSREFS
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KEYWORD
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base,nonn,new
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AUTHOR
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STATUS
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approved
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