cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A376045 Complement of A325112.

Original entry on oeis.org

3, 6, 9, 12, 13, 15, 16, 18, 19, 21, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 42, 43, 45, 46, 48, 49, 51, 53, 54, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 72, 73, 75, 76, 78, 79, 81, 83, 84, 86, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 102, 103, 105, 106, 108
Offset: 1

Views

Author

N. J. A. Sloane, Sep 09 2024

Keywords

Comments

Numbers missing from A325112.
Equivalently, numbers whose decimal expansion has a subsequence which is divisible by 3.

Links

Crossrefs

Cf. A325112.

Programs

  • Python
    from itertools import combinations
def A376045(n):
    def f(x):
        c, l = 0, len(str(x))
        for i in range(l):
            k = 10**i
            for j in (1,2,4,5,7,8):
                if j*k<=x:
                    c += 1
        for a in combinations((10**i for i in range(l)),2):
            for b in ((1, 1), (1, 4), (1, 7), (2, 2), (2, 5), (2, 8), (4, 1), (4, 4), (4, 7), (5, 2), (5, 5), (5, 8), (7, 1), (7, 4), (7, 7), (8, 2), (8, 5), (8, 8)):
                if a[0]*b[0]+a[1]*b[1] <= x:
                    c += 1
        return n+c
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Sep 10 2024

A325114 Integers k such that no nonzero subsequence of the decimal representation of k is divisible by 7.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 18, 19, 20, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33, 34, 36, 38, 39, 40, 41, 43, 44, 45, 46, 48, 50, 51, 52, 53, 54, 55, 58, 59, 60, 61, 62, 64, 65, 66, 68, 69, 80, 81, 82, 83, 85, 86, 88, 89, 90, 92, 93, 94, 95, 96, 99, 100, 101, 102, 103, 104, 106, 108, 109, 110, 111, 113, 115, 116, 118, 120
Offset: 1

Views

Author

Jonathan Kal-El Peréz, Mar 27 2019

Keywords

Comments

Does not contain 114 (helps to distinguish this from related sequences).
From David A. Corneth, Sep 10 2024: (Start)
Any term greater than 10^6 must have a digit 0. Proof: Any term between 10^6 and 10^7 has a 0.
Proof via induction and contradiction; any 7 digital number term has a digit 0. Suppose some number with k with q > 7 digits has no digit 0. Then floor(k/10) is a term and has no digit 0 and q - 1 digits. But there is no such number. A contradiction. Therefore any term with at least 7 digits has a digit 0. (End)

Links

Crossrefs

Cf. A014261 (for 2), A325112 (for 3), A325113 (for 4), A261189 (for 5).
See A376046 for complement.

Programs

  • Mathematica
    With[{k = 7}, Select[Range@ 100, NoneTrue[DeleteCases[FromDigits /@ Rest@ Subsequences[IntegerDigits@ #], 0], Mod[#, k] == 0 &] &]] (* Michael De Vlieger, Mar 31 2019 *)
  • PARI
    \\ See Corneth link

Extensions

More than the usual number of terms are shown in order to distinguish this from a new sequence arising from the game of "buzz" (cf. A092433). - N. J. A. Sloane, Sep 09 2024

A325113 Positive integers whose decimal representation has no nonzero subsequence that is divisible by 4.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 15, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 35, 37, 39, 50, 51, 53, 55, 57, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 75, 77, 79, 90, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 110, 111, 113, 115, 117
Offset: 1

Views

Author

Jonathan Kal-El Peréz, Mar 27 2019

Keywords

Comments

From Robert Israel, Apr 14 2020: (Start)
There are no digits 4 or 8.
If there is a digit 2 or 6, all previous digits must be even.
If there is a digit 0, all previous digits must be odd. (End)

Links

Crossrefs

Cf. A014261 (for 2), A325112 (for 3), A261189 (for 5).

Programs

  • Maple
    filter:= proc(n) local L,i;
  L:= convert(n,base,10);
  if member(4,L) or member(8,L) then return false fi;
  if member(0,L,i) and hastype(L[i+1..-1],even) then return false fi;
  i:= ListTools:-SelectFirst(t -> t=2 or t=6, L,output=indices);
  i = NULL or not hastype(L[i+1..-1],odd);
end proc:
select(filter, [$1..300]); # Robert Israel, Apr 14 2020
  • Mathematica
    With[{k = 4}, Select[Range@ 120, NoneTrue[DeleteCases[FromDigits /@ Rest@ Subsequences[IntegerDigits@ #], 0], Mod[#, k] == 0 &] &]] (* Michael De Vlieger, Mar 31 2019 *)

Extensions

Corrected by Robert Israel, Apr 14 2020

A330355 Starting from n: as long as the decimal representation contains a positive multiple of 3, divide the largest and leftmost such substring by 3; a(n) corresponds to the final value.

Original entry on oeis.org

0, 1, 2, 1, 4, 5, 2, 7, 8, 1, 10, 11, 4, 11, 14, 5, 4, 17, 2, 11, 20, 7, 22, 7, 8, 25, 22, 1, 28, 7, 10, 11, 4, 11, 14, 5, 4, 17, 2, 11, 40, 41, 14, 41, 44, 5, 14, 47, 4, 41, 50, 17, 52, 17, 2, 55, 52, 11, 58, 17, 20, 7, 22, 7, 8, 25, 22, 1, 28, 7, 70, 71, 8
Offset: 0

Views

Author

Rémy Sigrist, Dec 11 2019

Keywords

Comments

This sequence is a variant of A329424.

Examples

			For n = 193:
- 193 gives 1 followed by 93/3 = 131,
- 131 gives 1 followed by 3/3 followed by 1 = 111,
- 111 gives 111/3 = 37,
- 37 gives 3/3 followed by 7 = 17,
- neither 1, 7 nor 17 are divisible by 3, so a(193) = 17.

Links

Crossrefs

Cf. A325112, A327539, A329424.

Programs

  • Maple
    f:= proc(n) option remember; local L,m,i,d,np1,j,s;
  L:= convert(n,base,10);
  m:= nops(L);
  for d from m to 1 by -1 do
    for i from 1 to m-d+1 do
      s:= convert(L[i..i+d-1],`+`);
      if s > 0 and s mod 3 = 0 then
        np1:= add(L[j]*10^(j-1),j=1..i-1)+1/3*add(L[j]*10^(j-1),j=i..i+d-1);
        return procname(np1 + 10^(2+ilog10(np1)-(i+d))*add(L[j]*10^(j-1),j=i+d..m));
      fi
    od
  od;
  n
end proc:
map(f, [$0..100]); # Robert Israel, Dec 25 2019
  • PARI
    See Links section.

Formula

a(n) <= n with equality iff n = 0 or n belongs to A325112.
a(3^k) = 1 for any k >= 0.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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