A125289 -id:A125289 - cp's OEIS frontend

A061827 Number of partitions of n into parts which are the digits of n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 5, 4, 4, 3, 3, 3, 3, 1, 11, 1, 4, 7, 3, 5, 2, 4, 2, 1, 11, 6, 1, 3, 3, 7, 2, 2, 5, 1, 11, 11, 4, 1, 3, 4, 2, 7, 2, 1, 11, 6, 4, 3, 1, 2, 2, 2, 2, 1, 11, 11, 11, 6, 3, 1, 2, 3, 4, 1, 11, 6, 4, 3, 3, 2, 1, 2, 2, 1, 11, 11, 4, 11, 3, 4, 2, 1, 2, 1, 11, 6, 11, 3, 3, 6, 2, 2

Offset: 1

Amarnath Murthy, May 28 2001

a(A125289(n)) = 1, a(A125290(n)) > 1.

			For n = 11, 1+1+1+1+1+1+1+1+1+1+1. so a(11) = 1. For n = 12, 2+2+2+2+2+2 = 2+2+1+1+1+1+1+1+1+1 = ...etc
a(20) = 1: the only partitions permitted use the digits 0 and 2, so there is just 1, 20 = 2+2+2... ten times.

Alois P. Heinz, Table of n, a(n) for n = 1..15000 (first 1250 terms from Reinhard Zumkeller)

Cf. A061828, A109950, A119999, A125291, A136460, A193513.

import Data.List (sort, nub)
import Data.Char (digitToInt)
a061827 n =
   p n (map digitToInt $ nub $ sort $ filter (/= '0') $ show n) where
      p _ []        = 0
      p 0 _         = 1
      p m ds'@(d:ds)
        | m < d     = 0
        | otherwise = p (m - d) ds' + p m ds
-- Reinhard Zumkeller, Aug 01 2011

Length[IntegerPartitions[#,All,DeleteDuplicates@DeleteCases[IntegerDigits[#],0]]]&/@Range[200] (* Sander G. Huisman, Nov 14 2022 *)

More terms from David Wasserman, Jul 29 2002

A125292 Numbers having either no ones or no twos in their ternary representation.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 13, 18, 20, 24, 26, 27, 28, 30, 31, 36, 37, 39, 40, 54, 56, 60, 62, 72, 74, 78, 80, 81, 82, 84, 85, 90, 91, 93, 94, 108, 109, 111, 112, 117, 118, 120, 121, 162, 164, 168, 170, 180, 182, 186, 188, 216, 218, 222, 224, 234, 236, 240, 242, 243

Offset: 1

Reinhard Zumkeller, Nov 26 2006

Complement of A125293; union of A005823 and A005836;

A125291(a(n)) = 1; A062756(a(n))*A081603(a(n)) = 0.

Rémy Sigrist, Table of n, a(n) for n = 1..8190
Eric Weisstein's World of Mathematics, Ternary
Eric Weisstein's World of Mathematics, Repdigit

Subsequence of A154314.

Cf. A007089, A125289.

not[n_]:=Module[{c=DigitCount[n,3]},c[[1]]==0||c[[2]]==0]; Select[ Range[ 250],not] (* Harvey P. Dale, Dec 15 2012 *)

is(n, base=3) = #Set(select(sign, digits(n, base)))==1 \\ Rémy Sigrist, Mar 28 2020

a(n, base=3) = { for (w=0, oo, if (n<=(base-1)*2^w, my (d=1+(n-1)\2^w, k=2^w+(n-1)%(2^w)); return (d*fromdigits(binary(k), base)), n -= (base-1)*2^w)) } \\ Rémy Sigrist, Mar 28 2020

A336956 For any number n whose set of nonzero decimal digits is { d_0, ..., d_k } (with d_0 < ... < d_k), a(n) is obtained by replacing in the decimal representation of n each nonzero digit d_m by d_{k-m} for m = 0..k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 21, 31, 41, 51, 61, 71, 81, 91, 20, 12, 22, 32, 42, 52, 62, 72, 82, 92, 30, 13, 23, 33, 43, 53, 63, 73, 83, 93, 40, 14, 24, 34, 44, 54, 64, 74, 84, 94, 50, 15, 25, 35, 45, 55, 65, 75, 85, 95, 60, 16, 26, 36, 46, 56, 66, 76

Offset: 0

Rémy Sigrist, Aug 09 2020

This sequence is a self-inverse permutation of nonnegative integers.
This sequence first differs from A321474 for n = 112: a(112) = 221 whereas A321474(112) = 211.
This sequence has similarities with A166166; here we consider nonzero decimal digits, there binary run-lengths.

			For n = 10251:
- the set of nonzero digits is { 1, 2, 5},
- so we replace each digit 1, 2, 5 respectively by 5, 2, 1,
- and a(10251) = 50215.

Cf. A125289, A166166, A321474.

a(n, base=10) = { my (d=digits(n, base), s=Set(select(sign, d))); fromdigits(apply (t -> if (t, s[#s+1-setsearch (s,t)], 0), d), base) }

a(n) = n iff n = 0 or n belongs to A125289.

A382056 Remove every copy of the largest digit of n; if any digits remain, return the number formed by arranging the remaining digits in nondecreasing order. If no digits remain, return 0.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 2, 2, 2, 2, 2, 2, 0, 1, 2, 0, 3, 3, 3, 3, 3, 3, 0, 1, 2, 3, 0, 4, 4, 4, 4, 4, 0, 1, 2, 3, 4, 0, 5, 5, 5, 5, 0, 1, 2, 3, 4, 5, 0, 6, 6, 6, 0, 1, 2, 3, 4, 5, 6, 0, 7, 7, 0, 1, 2, 3, 4, 5, 6, 7, 0

Offset: 1

Ali Sada, Mar 13 2025

This sequence is inspired by a joke from Finnish comedian ISMO: "The hardest part in English for me is silent letters. You write them down but never say. But since you have silent letters in the language, why don’t you also have silent numbers? They would be far more useful. Like, I owe you 75 dollars, but the 7 is silent."

			For n = 132, we remove the largest digit and we get a(132) = 12. For n = 66, we remove the largest digit and we get a(66) = 0.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A054055 (largest digit of n), A125289 (numbers with only one (distinct) digit > 0), A382401.

a:= n-> (l-> (m-> (h-> add(h[j]*10^(j-1), j=1..nops(h))
     )(subs(m=NULL, l)))(max(l)))(convert(n, base, 10)):
seq(a(n), n=1..102);  # Alois P. Heinz, Mar 14 2025

A382056[n_] := FromDigits[DeleteCases[#, Max[#]]] & [IntegerDigits[n]];
Array[A382056, 100] (* Paolo Xausa, Mar 23 2025 *)

apply( {A382056(n, m=vecmax(digits(n)))=fromdigits([d | d<-digits(n), dM. F. Hasler, Mar 14 2025

def a(n):
    s = str(n)
    m = max(s)
    r = s.replace(m, "")
    return int(r) if r != "" else 0
print([a(n) for n in range(1, 103)]) # Michael S. Branicky, Mar 13 2025

a(n) = 0 for n in A125289. - M. F. Hasler, Mar 14 2025

Definition clarified by N. J. A. Sloane, Mar 23 2025

A333607 Numbers k with unique nonzero digit in decimal representation of 1/k.

Original entry on oeis.org

1, 2, 3, 5, 9, 10, 11, 15, 18, 20, 25, 30, 33, 45, 50, 90, 99, 100, 101, 110, 111, 125, 150, 165, 180, 198, 200, 225, 250, 300, 303, 330, 333, 450, 495, 500, 900, 909, 990, 999, 1000, 1001, 1010, 1100, 1110, 1111, 1125, 1250, 1287, 1500, 1515, 1650, 1665, 1800

Offset: 1

Rémy Sigrist, Mar 28 2020

This sequence has similarities with A125289; here we consider the decimal representation of 1/n, there that of n.
This sequence contains A333402.
If m belongs to the sequence, then 10*m also belongs to the sequence.

			The first terms, alongside their inverse, are:
  n   a(n)  1/a(n)
  --  ----  -----------
   1     1  1
   2     2  0.5
   3     3  0.333333...
   4     5  0.2
   5     9  0.111111...
   6    10  0.1
   7    11  0.090909...
   8    15  0.066666...
   9    18  0.055555...
  10    20  0.05
  11    25  0.04
  12    30  0.033333...
  13    33  0.030303...

Rémy Sigrist, PARI program for A333607

Cf. A125289, A333402.

PARI
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See Links section.
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A352057 Triangular numbers whose nonzero digits are all the same.

Original entry on oeis.org

0, 1, 3, 6, 10, 55, 66, 300, 666, 990, 3003, 5050, 10011, 66066, 500500, 600060, 50005000, 5000050000, 500000500000, 50000005000000, 5000000050000000, 500000000500000000, 50000000005000000000, 5000000000050000000000, 500000000000500000000000, 50000000000005000000000000

Offset: 1

Steven Lu, Mar 02 2022

This sequence may correspond to "monochromatic step squads" in the British animation "Numberblocks".

Conjecture: the largest term in this sequence whose nonzero digits are not 5 is 600060.

Supersequence of A037156.
Cf. A000217, A118668, A125289, A343811.
Cf. A352148 (indices of these triangular numbers).

(* Method1 *)
NonZeroQ[n_Integer] := n != 0; Select[
Table[n (n + 1)/2, {n, 0, 1000000}],
Length[Tally[Select[IntegerDigits[#], NonZeroQ]]] == 1 &]
(* Method2 *)
Sort[Select[
  Flatten[Outer[Times,
    Table[FromDigits[IntegerDigits[n, 2]], {n, 2^16 - 1}], Range[9]]],
   IntegerQ[Sqrt[8 # + 1]] &]]

isok(k) = my(d=digits(k*(k+1)/2)); d = select(x->(x!=0), d); #Set(d)<=1;
lista(nn) = {for (n=0, nn, if (isok(n), print1(n*(n+1)/2, ", ")););} \\ Michel Marcus, Mar 02 2022

from sympy import integer_nthroot
from sympy.utilities.iterables import multiset_permutations
def istri(n): return integer_nthroot(8*n+1, 2)[1]
def zplus1(digits):
    if digits == 1: yield 0
    for d1 in "123456789":
        digset = "0"*(digits-1) + d1*(digits-1)
        for mp in multiset_permutations(digset, digits-1):
            t = int(d1 + "".join(mp))
            yield t
def afind(maxdigits):
    for digits in range(1, maxdigits+1):
        for t in zplus1(digits):
            if istri(t):
                print(t, end=", ")
afind(22) # Michael S. Branicky, Mar 02 2022

a(24)-a(25) from Michael S. Branicky, Mar 02 2022

A343215 Differences between consecutive numbers with unique nonzero digit in decimal representation.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 2, 8, 3, 7, 4, 6, 5, 5, 6, 4, 7, 3, 8, 2, 9, 1, 1, 9, 1, 89, 2, 18, 2, 78, 3, 27, 3, 67, 4, 36, 4, 56, 5, 45, 5, 45, 6, 54, 6, 34, 7, 63, 7, 23, 8, 72, 8, 12, 9, 81, 9, 1, 1, 9, 1, 89, 1, 9, 1, 889, 2, 18, 2, 178, 2, 18, 2, 778

Offset: 1

Guillermo Miñones Collantes, Apr 08 2021

First differences of A125289.

a(n) = A125289(n+1) - A125289(n).

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A061827 Number of partitions of n into parts which are the digits of n.

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A125292 Numbers having either no ones or no twos in their ternary representation.

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A336956 For any number n whose set of nonzero decimal digits is { d_0, ..., d_k } (with d_0 < ... < d_k), a(n) is obtained by replacing in the decimal representation of n each nonzero digit d_m by d_{k-m} for m = 0..k.

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A382056 Remove every copy of the largest digit of n; if any digits remain, return the number formed by arranging the remaining digits in nondecreasing order. If no digits remain, return 0.

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A333607 Numbers k with unique nonzero digit in decimal representation of 1/k.

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A352057 Triangular numbers whose nonzero digits are all the same.

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A343215 Differences between consecutive numbers with unique nonzero digit in decimal representation.

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