A129459 -id:A129459 - cp's OEIS frontend

A375211 Complement of A129459.

3, 4, 5, 7, 9, 16, 18, 19, 23, 25, 33, 34, 38, 43, 45, 48, 49, 54, 56, 58, 59, 66, 77, 79, 84, 86, 93, 144, 149, 155, 157, 166, 176, 189, 195, 223, 233, 238, 243, 258, 266, 293, 295, 304, 308, 313, 314, 323, 333, 334, 344, 376, 378, 388, 393, 404, 423, 433, 435, 443, 448, 457, 459, 566, 576, 579

Offset: 1

Robert Israel, Feb 06 2025

Given positive integer x, let y be the greatest integer < x that is not in the sequence (and thus is a member of A129459). Then x is in the sequence iff neither x nor y share any decimal digit with x * y.

For each k >= 1, at least one of (10^k - 1)/3 = 3...33 and (10^k + 2)/3 = 3...34 is in the sequence, as their product is (10^(2*k) + 10^k - 2)/9 = 1..12...2. Similarly, at least one of (2*10^k - 2)/3 = 6...66 and (2*10^k + 1)/3 = 6...67 is in the sequence. In particular, the sequence is infinite.

			a(6) = 16 is a term because 15 is the greatest integer < 16 that is not in the sequence, and neither 16 nor 15 shares a digit with 16 * 15 = 240.

Robert Israel, Table of n, a(n) for n = 1..5000

Cf. A129459.

f:= proc(n) local Ln, Lk,k;
  Ln:= convert(convert(n,base,10),set);
  for k from n+1 do
     Lk:= convert(convert(k,base,10),set) union Ln;
     if convert(convert(n*k,base,10),set) intersect Lk <> {} then return k fi
  od
end proc:
R:= NULL: x:= 0: count:= 0:
while count < 100 do
  y:= f(x);
  count:= count + y - x - 1;
  R:= R, $(x+1)..(y-1);
  x:= y
od:
R;

A129460 Third column (m=2) of triangle A129065.

Original entry on oeis.org

1, 10, 156, 3696, 125280, 5780160, 349090560, 26760222720, 2540101939200, 292579402752000, 40213832085504000, 6502800338141184000, 1222285449585328128000, 264279998869470904320000

Offset: 0

Wolfdieter Lang, May 04 2007

See A129065 for the M. Bruschi et al. reference.

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A129065, A129459 (m=1), A129461 (m=3).

function T(n,k) // T = A129065
  if k lt 0 or k gt n then return 0;
  elif n eq 0 then return 1;
  else return 2*(n-1)^2*T(n-1,k) - 4*Binomial(n-1,2)^2*T(n-2,k) + T(n-1,k-1);
  end if;
end function;
A129460:= func< n | T(n+2, 2) >;
[A129460(n): n in [0..20]]; // G. C. Greubel, Feb 08 2024

T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n==0, 1, 2*(n-1)^2*T[n-1,k] - 4*Binomial[n-1,2]^2*T[n-2,k] +T[n-1,k-1] ]]; (* T=A129065 *)
A129460[n_]:= T[n+2,2];
Table[A129460[n], {n,0,40}] (* G. C. Greubel, Feb 08 2024 *)

@CachedFunction
def T(n,k): # T = A129065
    if (k<0 or k>n): return 0
    elif (n==0): return 1
    else: return 2*(n-1)^2*T(n-1,k) - 4*binomial(n-1,2)^2*T(n-2,k) + T(n-1,k-1)
def A129460(n): return T(n+2,2)
[A129460(n) for n in range(41)] # G. C. Greubel, Feb 08 2024

a(n) = A129065(n+2, 2), n >= 0.

A129513 Slowest increasing sequence that starts with 2 and has property that multiplying two consecutive terms gives a number which does not share a digit with either of the two terms.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 42, 43, 44, 45, 46, 63, 66, 67, 73, 76, 77, 78, 79, 84, 88, 113, 183, 219, 232, 233, 236, 333, 334, 335, 522, 577, 579, 589, 698, 738, 803, 819, 858, 859, 899, 3033, 3037, 3127, 12792, 27117, 29662, 29669, 29777, 33953, 34409, 34474

Offset: 1

Eric Angelini, May 29 2007

Terms computed by Stefan Steinerberger. Is the sequence finite?

Andrew Woods, Table of n, a(n) for n = 1..175

Cf. A129459.

cp's OEIS Frontend

A375211 Complement of A129459.

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A129460 Third column (m=2) of triangle A129065.

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A129513 Slowest increasing sequence that starts with 2 and has property that multiplying two consecutive terms gives a number which does not share a digit with either of the two terms.

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