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a(n) = smallest positive k such that scan_diff(k,n) is a square, where scan_diff is defined in the Comments.
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%I #11 Dec 08 2019 07:39:01

%S 2,1,2,3,1,2,3,4,5,3,1,2,3,1,2,3,4,5,1,2,3,4,1,2,3,4,5,1,2,1,2,1,2,3,

%T 4,5,1,2,3,5,1,2,3,4,5,1,2,3,4,4,5,6,7,8,1,2,3,4,5,3,4,5,6,1,2,3,4,5,

%U 6,2,3,4,1,2,3,4,5,6,7,1,2,1,2,3,4,5,6

%N a(n) = smallest positive k such that scan_diff(k,n) is a square, where scan_diff is defined in the Comments.

%C Write i, j in base 10 aligned to right, say

%C i = bcd...ef

%C j = .gh...pq

%C Then scan_diff(i,j) = |b-0| + |c-g| + |d-h| + ... + |e-p| + |f-q|.

%C Example: scan_diff(12345,909) = 1+2+6+4+4 = 17.

%C Suggested by the definition of "box" in A329794.

%H Rémy Sigrist, <a href="/A329795/b329795.txt">Table of n, a(n) for n = 1..25000</a>

%e For n = 1 the smallest k producing a square is 2 (as scan_diff(1,2) = 1);

%e For n = 2 the smallest k producing a square is 1 (as scan_diff(2,1) = 1);

%e For n = 3 the smallest k producing a square is 2 (as scan_diff(3,2) = 1);

%e For n = 5 the smallest k producing a square is 1 (as scan_diff(5,1) = 4);

%e For n = 16 the smallest k producing a square is 3 (as scan_diff(16,3) = 1+3 = 4).

%o (PARI) scan_diff(n,k) = if (n*k, scan_diff(n\10,k\10)+abs((n%10)-(k%10)), n+k)

%o a(n) = for (k=1, oo, my (t=scan_diff(n,k)); if (t && issquare(t), return (k))) \\ _Rémy Sigrist_, Dec 08 2019

%Y Cf. A329794.

%K nonn,base

%O 1,1

%A _N. J. A. Sloane_, Dec 07 2019

%E More terms from _Rémy Sigrist_, Dec 08 2019