A359736 - OEIS

%I #7 Feb 04 2023 16:39:15

%S 0,10,1,2,6,42,20,7,11,4,56,3,5,21,30,43,12,31,14,8,13,9,29,19,15,18,

%T 22,17,24,32,72,26,28,90,23,91,35,109,37,41,48,73,27,34,50,57,38,40,

%U 33,47,71,51,62,55,66,89,112,16,79,39,130,63,46,44,65,25,135

%N Lexicographically earliest sequence of distinct nonnegative integers such that the sequence d(n) = dist(a(n), SQUARES) has the same sequence of digits.

%C In the definition, dist(a(n), SQUARES) = A053188(a(n)) is the distance of a(n) from the nearest square. "... has the same digits" means that the concatenation of the terms yields the same string of digits, for the sequence a(.) and the sequence d(.).

%C Conjectured to be a permutation of the nonnegative integers. The inverse permutation would start (0, 2, 3, 11, 9, 12, 4, 7, 19, 21, 1, 8, 16, 20, 18, 24, ...)

%H Eric Angelini, <a href="https://cinquantesignes.blogspot.com/2023/01/digit-spines.html">Digit-spines</a>, personal blog "Cinquante signes" on blogspot.com, Jan. 11, 2023.

%e Below, row "s" lists the closest square to a(n) and row "d" the absolute difference |a(n)-s|. We have the same sequence of digits in rows a (this sequence) and d:

%e   n :  0   1   2   3   4   5   6   7   8   9  10  11  12  13  14 ...

%e   a :  0  10   1   2   6  42  20   7  11   4  56   3   5  21  30 ...

%e   s :  0   9   1   1   4  36  16   9   9   4  49   4   6  25  25 ...

%e   d :  0   1   0   1   2   6   4   2   2   0   7   1   1   4   5 ...

%o (PARI) spine(x->x^2, 200) \\ See A359734 for spine()

%Y Cf. A053188 (distance from the nearest square), A000290 (the squares).

%Y Cf. A359734, A359737 (similar for primes and Fibonacci numbers).

%K nonn,base

%O 0,2

%A _M. F. Hasler_ and _Eric Angelini_, Jan 12 2023