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A265650
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Removing the first occurrence of 1, 2, 3, ... reproduces the sequence itself. Each run of consecutive removed terms is separated from the next one by a term a(k) <= a(k-1) such that floor(sqrt(a(k))) equals the length of the run.
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4
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1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 8, 4, 9, 1, 10, 11, 5, 12, 3, 13, 14, 6, 15, 2, 16, 17, 7, 18, 19, 8, 20, 21, 4, 22, 23, 24, 9, 25, 1, 26, 27, 28, 10, 29, 30, 31, 11, 32, 33, 5, 34, 35, 36, 12, 37, 3, 38, 39, 40, 13, 41, 42, 43, 14, 44, 45, 6, 46, 47, 48, 15, 49, 2, 50, 51, 52, 53, 16, 54, 55, 56, 57, 17, 58, 59, 7, 60, 61, 62, 63, 18, 64, 65, 66
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OFFSET
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1,3
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COMMENTS
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A fractal sequence : If one deletes the first occurrence of 1, 2, 3, ... the original sequence is reproduced.
Subsequent runs of consecutive terms which are these first occurrences are separated by a term whose square root yields the length of the preceding run (when rounded down).
Motivated by Project Euler problem 535, see LINKS.
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LINKS
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FORMULA
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The sequence contains marked numbers and non-marked numbers.
The marked numbers are consecutive starting with a(1)=1.
Immediately preceding each non-marked number in a(n), there are exactly floor(sqrt(a(n)) [= A000196(a(n))] adjacent marked numbers.
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EXAMPLE
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The runs of first occurrences of the positive integers are {1}, {2}, {3}, {4}, {5}, {6}, {7, 8}, {9}, {10, 11}, ... each separated from the next one by, respectively, 1, 1, 2, 1, 3, 2, 4, 1, 5, ... where 4 and 5 follow the groups {7, 8} and {10, 11} of length 2 = sqrt(4) = floor(sqrt(5)). - M. F. Hasler, Dec 13 2015
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PROG
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(C)
#include <stdio.h>
#include <math.h>
#define SIZE 1000
unsigned int numbers[SIZE];
int main() {
unsigned int pointer=0, next=1, circle_count=1, next_circle_number=2, sqrt_non_circle=1;
numbers[0]=1; printf("1");
while (next<SIZE) {
if (circle_count==sqrt_non_circle) {
numbers[next]=numbers[pointer]; circle_count=0; pointer++;
sqrt_non_circle=sqrt(numbers[pointer]);
} else {
circle_count++; numbers[next]=next_circle_number;
next_circle_number++;
}
printf(", %u", numbers[next]); next++;
}
}
(PARI) A265650(n, list=0, a=[1], cc=0, nc=1, p=0)={for(i=2, n, a=concat(a, if(0<=cc-=1, nc+=1, cc=sqrtint(a[!!p+p+=1]); a[p]))); list&&return(a); a[n]} \\ Set 2nd optional arg.to 1 to return the whole list. - M. F. Hasler, Dec 13 2015
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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