A265650 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.

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

Offset

1,3

Comments

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.

The sequence contains marked numbers and non-marked numbers. A number is marked if it is the first occurrence of that number.

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.

Links

Martin Møller Skarbiniks Pedersen, Table of n, a(n) for n = 1..1000

Project Euler, Problem 535: Fractal Sequence

Clark Kimberling, Interspersions and Fractal Sequences Associated with Fractions c^j/d^k, Journal of Integer Sequences, Issue 5, Volume 10 (2007), Article 07.5.1

Example

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

Prog

(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

Crossrefs

Cf. A000196, A003603, A035513.

Sequence in context: A366893 A365388 A366380 * A181733 A049773 A261401

Adjacent sequences: A265647 A265648 A265649 * A265651 A265652 A265653

Keyword

easy,nonn,changed

Author

Martin Møller Skarbiniks Pedersen, Dec 11 2015

Extensions

New name from M. F. Hasler, Dec 13 2015

Status

approved