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
Examples
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).
Links
- Rémy Sigrist, Table of n, a(n) for n = 1..25000
Programs
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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 *) -
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
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PARI
A329794(n)={n>1&&for(k=1,n,issquare(A330240(n,k))&&return(k));2} \\ M. F. Hasler, Dec 07 2019
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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
Formula
a(n) < n except for a(1) = 2. - M. F. Hasler, Dec 07 2019
Comments