A329794 a(n) is the smallest positive k such that box(k,n) is a positive square, where box(k,n) is Eric Angelini's mapping defined in the Comments.

2, 1, 2, 3, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 5, 6, 7, 8, 9, 21, 1, 2, 3, 4, 6, 7, 8, 9, 19, 10, 11, 1, 2, 3, 9, 17, 18, 19, 29, 20, 10, 11, 12, 13, 19, 27, 28, 29, 39, 30, 20, 21, 22, 10, 4, 5, 6, 7, 8, 1

Offset

1,1

Comments

Eric Angelini's "box" map box(i,j) is defined as follows (see A330240). Write i, j in base 10 aligned to the right, say

i = bcd...ef

j = .gh...pq

Then the decimal expansion of box(i,j) is |b-0|, |c-g|, |d-h|, ..., |e-p|, |f-q|.

For example, box(12345,909) = 12644.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..25000

Formula

a(n) < n except for a(1) = 2. - M. F. Hasler, Dec 07 2019

Example

For n = 1 the smallest k producing a square is 2 (as box(1,2) = 1, this 1 being the square of 1);
For n = 2 the smallest k producing a square is 1 (as box(2,1) = 1, this 1 being the square of 1);
For n = 3 the smallest k producing a square is 2 (as box(3,2) = 1, this 1 being the square of 1);
For n = 5 the smallest k producing a square is 3 (as box(5,1) = 4, this 4 being the square of 2);
For n = 16 the smallest k producing a square is 12 (as box(16,12) = 4, this 4 being the square of 2).

Mathematica

BOX[a_, b_]:=FromDigits@Abs[Subtract@@PadLeft[IntegerDigits/@{a, b}]]; Table[k=1; While[!IntegerQ[a=Sqrt@BOX[k, n]]||a==0, k++]; k, {n, 100}] (* Giorgos Kalogeropoulos, Aug 20 2021 *)

Prog

(PARI) box(x, y) = if (x==0 || y==0, x+y, 10*box(x\10, y\10) + abs((x%10) - (y%10)))
a(n) = for (k=1, oo, my (b=box(n, k)); if (b && issquare(b), return (b))) \\ Rémy Sigrist, Dec 07 2019
(PARI) A329794(n)={n>1&&for(k=1, n, issquare(A330240(n, k))&&return(k)); 2} \\ M. F. Hasler, Dec 07 2019
(Python)
from sympy.ntheory.primetest import is_square
def positive_square(n): return n > 0 and is_square(n)
def box(i, j):
    si = str(i); sj = str(j); m = max(len(si), len(sj))
    si, sj = si.zfill(m), sj.zfill(m)
    return int("".join([str(abs(int(si[k])-int(sj[k]))) for k in range(m)]))
def a(n):
    k = 1
    while not positive_square(box(k, n)): k += 1
    return k
print([a(n) for n in range(1, 66)]) # Michael S. Branicky, Aug 20 2021

Crossrefs

Cf. A329795, A330240.

Sequence in context: A194055 A162192 A329795 * A238954 A238967 A066657

Adjacent sequences: A329791 A329792 A329793 * A329795 A329796 A329797

Keyword

nonn,base,look

Author

N. J. A. Sloane, Dec 07 2019, based on a posting by Eric Angelini to the Sequence Fans Mailing List, Dec 07 2019. (Thanks to Rémy Sigrist for correcting the definition.)

Status

approved