A330240 -id:A330240 - cp's OEIS frontend

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

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.)

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.

			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).

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

Cf. A329795, A330240.

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 *)

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

A329794(n)={n>1&&for(k=1,n,issquare(A330240(n,k))&&return(k));2} \\ M. F. Hasler, Dec 07 2019

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

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

A330237 Square array T(n,k): concatenate the absolute differences of the digits of n and k (the smaller one padded with leading zeros); read by antidiagonals; n, k >= 1.

Original entry on oeis.org

0, 1, 1, 2, 0, 2, 3, 1, 1, 3, 4, 2, 0, 2, 4, 5, 3, 1, 1, 3, 5, 6, 4, 2, 0, 2, 4, 6, 7, 5, 3, 1, 1, 3, 5, 7, 8, 6, 4, 2, 0, 2, 4, 6, 8, 11, 7, 5, 3, 1, 1, 3, 5, 7, 11, 10, 12, 6, 4, 2, 0, 2, 4, 6, 12, 10, 11, 11, 13, 5, 3, 1, 1, 3, 5, 13, 11, 11, 12, 10, 12, 14, 4, 2, 0, 2, 4, 14, 12, 10, 12, 13, 11, 11, 13, 15, 3, 1, 1, 3, 15, 13, 11, 11, 13, 14, 12, 10, 12, 14, 16, 2, 0, 2, 16

Offset: 1

M. F. Hasler, Dec 06 2019

A digit-wise analog of A049581.

The binary operator T: N x N -> N is commutative, therefore this table is symmetric and it does not matter in which direction the antidiagonals are read. It would also be sufficient to specify only the lower half of the square table: see A330238 for this variant. The operator is also defined for either argument equal to 0, which is the neutral element: T(x,0) = 0 for all x. Therefore we omit row & column 0 here, see A330240 for the table including these. Every element is its opposite or inverse, as shown by the zero diagonal T(x,x) = 0.

			The square array starts as follows:
   n | k=1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 ...
  ---+-----------------------------------------------------------
   1 |   0  1  2  3  4  5  6  7  8 11 10 11 12 13 14 15 16 17 ...
   2 |   1  0  1  2  3  4  5  6  7 12 11 10 11 12 13 14 15 16 ...
   3 |   2  1  0  1  2  3  4  5  6 13 12 11 10 11 12 13 14 15 ...
   4 |   3  2  1  0  1  2  3  4  5 14 13 12 11 10 11 12 13 14 ...
   5 |   4  3  2  1  0  1  2  3  4 15 14 13 12 11 10 11 12 13 ...
   6 |   5  4  3  2  1  0  1  2  3 16 15 14 13 12 11 10 11 12 ...
   7 |   6  5  4  3  2  1  0  1  2 17 16 15 14 13 12 11 10 11 ...
   8 |   7  6  5  4  3  2  1  0  1 18 17 16 15 14 13 12 11 10 ...
   9 |   8  7  6  5  4  3  2  1  0 19 18 17 16 15 14 13 12 11 ...
  10 |  11 12 13 14 15 16 17 18 19  0  1  2  3  4  5  6  7  8 ...
  11 |  10 11 12 13 14 15 16 17 18  1  0  1  2  3  4  5  6  7 ...
  12 |  11 10 11 12 13 14 15 16 17  2  1  0  1  2  3  4  5  6 ...
   (...)
It differs from A049581 only if at least one index is > 10.

Eric Angelini, The box ■ operation, personal blog "Cinquante signes", and post to the SeqFan list, Dec 06 2019.
Eric Angelini, The box ■ operation, personal blog "Cinquante signes", and post to the SeqFan list, Dec 06 2019. [Cached copy]

Cf A330240 (variant including row & column 0), A330237 (lower left triangle), A049581 (T(n,k) = |n-k|).

T(a,b)=fromdigits(abs(Vec(digits(min(a,b)),-logint(a=max(a,b),10)-1)-digits(a)))

A330238 Triangle T(n,k): concatenate the absolute differences of the digits of n and k (the smaller one padded with leading zeros); n >= k >= 1.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 4, 3, 2, 1, 0, 5, 4, 3, 2, 1, 0, 6, 5, 4, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 10, 11, 12, 13, 14, 15, 16, 17, 18, 1, 0, 11, 10, 11, 12, 13, 14, 15, 16, 17, 2, 1, 0, 12, 11, 10, 11, 12, 13, 14, 15, 16, 3, 2, 1, 0, 13, 12, 11, 10, 11, 12, 13, 14, 15, 4, 3, 2, 1, 0

Offset: 1

M. F. Hasler, Dec 06 2019

A digit-wise analog of A049581.

The binary operator T: N x N -> N is commutative, so we need only the lower half of the symmetric square table A330238 or A330240 (including n, k = 0). Also, 0 is the neutral element: T(x,0) = x for all x, therefore we omit row & column 0. The trivial diagonal T(x,x) = 0 could also be omitted but serves as an end-of-row marker and makes indexing simpler and more natural.

			The triangle starts as follows:
    n | k=1  2   3   4   5   6   7   8   9  10  11
   ---+-------------------------------------------
    1 |  0,
    2 |  1,  0,
    3 |  2,  1,  0,
    4 |  3,  2,  1,  0,
    5 |  4,  3,  2,  1,  0,
    6 |  5,  4,  3,  2,  1,  0,
    7 |  6,  5,  4,  3,  2,  1,  0,
    8 |  7,  6,  5,  4,  3,  2,  1,  0,
    9 |  8,  7,  6,  5,  4,  3,  2,  1,  0,
   10 | 11, 12, 13, 14, 15, 16, 17, 18, 19,  0,
   11 | 10, 11, 12, 13, 14, 15, 16, 17, 18,  1,  0,
   12 | 11, 10, 11, 12, 13, 14, 15, 16, 17,  2,  1,  0,
    (...)

Eric Angelini, The box ■ operation, personal blog "Cinquante signes", and post to the SeqFan list, Dec 06 2019.
Eric Angelini, The box ■ operation, personal blog "Cinquante signes", and post to the SeqFan list, Dec 06 2019. [Cached copy]

Cf. A330237 (same as a square array read by antidiagonals), A330240 (idem, including row & column 0), A049581 (T(n,k) = |n-k|).

A330238(n,k)=fromdigits(digits(n)-abs(Vec(digits(k),-logint(n,10)-1))) \\ see A330240 for a more general function not limited to 1 <= k <= n

A338827 For any number with decimal representation (d(1), d(2), ..., d(k)), the decimal representation of a(n) is (abs(d(1)-d(k)), abs(d(2)-d(k-1)), ..., abs(d(k)-d(1))).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 11, 22, 33, 44, 55, 66, 77, 88, 22, 11, 0, 11, 22, 33, 44, 55, 66, 77, 33, 22, 11, 0, 11, 22, 33, 44, 55, 66, 44, 33, 22, 11, 0, 11, 22, 33, 44, 55, 55, 44, 33, 22, 11, 0, 11, 22, 33, 44, 66, 55, 44, 33, 22, 11, 0, 11, 22

Offset: 0

Rémy Sigrist, Nov 11 2020

Leading zeros are ignored.

All terms belong to A061917.

			For n = 1021:
- abs(1-1) = 0,
- abs(0-2) = 2,
- abs(2-0) = 2,
- abs(1-1) = 0,
- so a(1021) = 220.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A002113, A004086, A056965, A061917, A175919 (binary analog), A330240, A338828 (ternary analog).

a:= n-> (l-> (h-> add(h[j]*10^(j-1), j=1..nops(h)))([seq(
    abs(l[i]-l[-i]), i=1..nops(l))]))(convert(n, base, 10)):
seq(a(n), n=0..70);  # Alois P. Heinz, Nov 12 2020

a(n, base=10) = my (d=digits(n, base)); fromdigits(abs(d-Vecrev(d)), base)

a(n) = 0 iff n is a palindrome (A002113).

a(n) = A330240(n, A004086(n)).

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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.

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A330237 Square array T(n,k): concatenate the absolute differences of the digits of n and k (the smaller one padded with leading zeros); read by antidiagonals; n, k >= 1.

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A330238 Triangle T(n,k): concatenate the absolute differences of the digits of n and k (the smaller one padded with leading zeros); n >= k >= 1.

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A338827 For any number with decimal representation (d(1), d(2), ..., d(k)), the decimal representation of a(n) is (abs(d(1)-d(k)), abs(d(2)-d(k-1)), ..., abs(d(k)-d(1))).

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