A121042 -id:A121042 - cp's OEIS frontend

A011531 Numbers that contain a digit 1 in their decimal representation.

1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 31, 41, 51, 61, 71, 81, 91, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133

Offset: 1

N. J. A. Sloane

A121042(a(n)) = 1. - Reinhard Zumkeller, Jul 21 2006

See A043493 for numbers that contain a single digit '1'. A subsequence of numbers having a digit that divides all other digits, A263314. - M. F. Hasler, Jan 11 2016

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Cf. A121041, A121022, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040.

Cf. A052383 (complement), A062634 (a subsequence).

Filtered([1..140],n->1 in ListOfDigits(n)); # Muniru A Asiru, Feb 23 2019

a011531 n = a011531_list !! (n-1)
a011531_list = filter ((elem '1') . show) [0..]
-- Reinhard Zumkeller, Feb 05 2012

[n: n in [0..500] | 1 in Intseq(n) ]; // Vincenzo Librandi, Jan 11 2016

M:= 3: # to get all terms of up to M digits
B:= {1}: A:= {1}:
for i from 2 to M do
   B:= map(t -> seq(10*t+j,j=0..9),B) union
      {seq(10*x+1,x=2*10^(i-2)..10^(i-1)-1)}:
   A:= A union B;
od:
sort(convert(A,list)); # Robert Israel, Jan 10 2016
# second program:
A011531 := proc(n)
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if nops(convert(convert(a,base,10),set) intersect {1}) > 0 then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Jul 31 2016

Select[Range[600] - 1, DigitCount[#, 10, 1] > 0 &] (* Vincenzo Librandi, Jan 11 2016 *)

is_A011531(n)=setsearch(Set(digits(n)),1) \\ M. F. Hasler, Jan 10 2016

def aupto(nn): return [m for m in range(1, nn+1) if '1' in str(m)]
print(aupto(133)) # Michael S. Branicky, Jan 10 2021

(0 to 119).filter(.toString.indexOf('1') > -1) // _Alonso del Arte, Jan 12 2020

a(n) ~ n. - Charles R Greathouse IV, Nov 02 2022

A121041 Number of divisors of n that are also contained in the decimal representation of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 1, 3, 2, 2, 1, 2, 1, 2, 2, 2, 2, 1, 2, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 1, 3, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 2, 1, 2, 2, 1, 1, 2, 3, 2, 3, 2, 3, 3

Offset: 1

Reinhard Zumkeller, Jul 21 2006

			a(22) = #{2, 22} = 2;
a(23) = #{23} = 1;
a(24) = #{2, 4, 24} = 3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A121022, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040.

import Data.List (isInfixOf)
a121041 n = length $ filter (\d -> n `mod` d == 0
                                   && show d `isInfixOf` show n) [1..n]
-- Reinhard Zumkeller, Feb 11 2011

A121041[n_] := DivisorSum[n, 1 &, StringContainsQ[IntegerString[n], IntegerString[#]] &]; Array[A121041, 150] (* Paolo Xausa, Feb 25 2024 *)

substr(a,b)=a=digits(a); b=digits(b); for(i=0,#a-#b, for(j=1,#b, if(a[i+j]!=b[j], next(2))); return(1)); 0
a(n)=sumdiv(n,d, substr(n,d)) \\ Charles R Greathouse IV, Mar 31 2016

from sympy import divisors
def a(n):
    s = str(n)
    return sum(1 for d in divisors(n, generator=True) if str(d) in s)
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Jul 11 2022

a(n) = 1 iff A121042(n) = n.

a(A155005(n)) = n and a(m) < n for m < A155005(n). - Reinhard Zumkeller, Jan 18 2009

A355620 a(n) is the sum of the divisors of n whose decimal expansions appear as substrings in the decimal expansion of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 14, 15, 21, 17, 18, 19, 20, 22, 22, 24, 23, 30, 30, 28, 27, 30, 29, 33, 32, 34, 36, 34, 40, 45, 37, 38, 42, 44, 42, 44, 43, 48, 50, 46, 47, 60, 49, 55, 52, 54, 53, 54, 60, 56, 57, 58, 59, 66, 62, 64, 66, 68, 70, 72, 67

Offset: 1

Rémy Sigrist, Jul 10 2022

			For n = 110:
- the divisors of 110 are: 1, 2, 5, 10, 11, 22, 55, 110,
- 1, 10, 11 and 110 appear as substrings in 110,
- so a(110) = 1 + 10 + 11 + 110 = 132.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of the first 100000 terms (red pixels indicate when n is a multiple of 10)
Index entries for sequences related to decimal expansion of n
Index entries for sequences related to divisors

Cf. A000203, A002275, A121041, A121042, A239058, A355633 (binary analog).

Table[DivisorSum[n, # &, StringContainsQ[IntegerString[n], IntegerString[#]] &], {n, 100}] (* Paolo Xausa, Jul 23 2024 *)

a(n, base=10) = { my (d=digits(n, base), s=setbinop((i,j) -> fromdigits(d[i..j], base), [1..#d]), v=0); for (k=1, #s, if (s[k] && n%s[k]==0, v+=s[k])); return (v) }

from sympy import divisors
def a(n):
    s = str(n)
    return sum(d for d in divisors(n, generator=True) if str(d) in s)
print([a(n) for n in range(1, 68)]) # Michael S. Branicky, Jul 10 2022

a(n) >= n.
a(n) <= A000203(n) with equality iff n belongs to A239058.
a(10^n) = A002275(n+1) for any n >= 0.

A355632 Irregular triangle T(n, k), n > 0, k = 1..A121041(n), read by rows; the n-th row contains in ascending order the divisors of n whose decimal expansions appear as substrings in the decimal expansion of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 10, 1, 11, 1, 2, 12, 1, 13, 1, 14, 1, 5, 15, 1, 16, 1, 17, 1, 18, 1, 19, 2, 20, 1, 21, 2, 22, 23, 2, 4, 24, 5, 25, 2, 26, 27, 2, 28, 29, 3, 30, 1, 31, 2, 32, 3, 33, 34, 5, 35, 3, 6, 36, 37, 38, 3, 39, 4, 40, 1, 41, 2, 42, 43, 4, 44

Offset: 1

Rémy Sigrist, Jul 11 2022

			Triangle T(n, k) begins:
     1: [1]
     2: [2]
     3: [3]
     4: [4]
     5: [5]
     6: [6]
     7: [7]
     8: [8]
     9: [9]
    10: [1, 10]
    11: [1, 11]
    12: [1, 2, 12]
    13: [1, 13]
    14: [1, 14]
    15: [1, 5, 15]
    16: [1, 16]

Paolo Xausa, Table of n, a(n) for n = 1..12288 (rows 1..5000 of the triangle, flattened).
Index entries for sequences related to decimal expansion of n
Index entries for sequences related to divisors

Cf. A027750, A121041 (row lengths), A121042, A355620 (row sums), A355634 (binary analog).

Table[Select[Divisors[n], StringContainsQ[IntegerString[n], IntegerString[#]] &], {n, 50}] (* Paolo Xausa, Jul 23 2024 *)

row(n, base=10) = { my (d=digits(n, base), s=setbinop((i,j) -> fromdigits(d[i..j], base), [1..#d]), v=0); select(v -> v && n%v==0, s) }

from sympy import divisors
def row(n):
    s = str(n)
    return sorted(d for d in divisors(n, generator=True) if str(d) in s)
def table(r): return [i for n in range(1, r+1) for i in row(n)]
print(table(44)) # Michael S. Branicky, Jul 11 2022

T(n, 1) = A121042(n).
T(n, A121041(n)) = n.
Sum_{k = 1..A121041(n)} T(n, k) = A355620(n).

A383749 Positive numbers k whose decimal expansion does not contain the decimal expansion of any proper divisor of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 23, 27, 29, 34, 37, 38, 43, 46, 47, 49, 53, 54, 56, 57, 58, 59, 67, 68, 69, 73, 74, 76, 78, 79, 83, 86, 87, 89, 94, 97, 98, 203, 207, 209, 223, 227, 229, 233, 239, 247, 249, 253, 257, 259, 263, 267, 269, 277, 283, 289, 293, 299, 307

Offset: 1

Rémy Sigrist, May 08 2025

Also fixed points of A121042.
This sequence is infinite as it contains A173041.
a(n) > 5 contains no decimal digit 1 and does not end in 2 or 5. - Michael S. Branicky, May 11 2025

			The proper divisors of 54 are 1, 2, 3, 6, 9, 18 and 27; none of them appear in the decimal expansion of 54 so 54 belongs to this sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A011531, A027751, A038772, A121042, A383592.

A038603 and A173041 are subsequences.

A383749Q[k_] := SelectFirst[Divisors[k], StringContainsQ[IntegerString[k], IntegerString[#]] &] == k;
Select[Range[500], A383749Q] (* Paolo Xausa, May 12 2025 *)

is(n, base = 10) = {
    my (d = digits(n, base));
    for (i = 1, #d,
        if (d[i],
            for (j = i, #d,
                if ((i!=1 || j!=#d) && n % fromdigits(d[i..j], base)==0,
                    return (0);););););
    return (1);}

from itertools import count, islice
from sympy import divisors
def A383749_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:not any(dA383749_list = list(islice(A383749_gen(),40)) # Chai Wah Wu, May 10 2025

def ok(n):
    s = str(n)
    subs = (s[i:j] for i in range(len(s)) for j in range(i+1, len(s)+1) if s[i]!='0')
    return n and not any(n%v == 0 for ss in subs if n > (v:=int(ss)) > 0)
print([k for k in range(308) if ok(k)]) # Michael S. Branicky, May 11 2025

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A011531 Numbers that contain a digit 1 in their decimal representation.

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A121041 Number of divisors of n that are also contained in the decimal representation of n.

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A355620 a(n) is the sum of the divisors of n whose decimal expansions appear as substrings in the decimal expansion of n.

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A355632 Irregular triangle T(n, k), n > 0, k = 1..A121041(n), read by rows; the n-th row contains in ascending order the divisors of n whose decimal expansions appear as substrings in the decimal expansion of n.

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A383749 Positive numbers k whose decimal expansion does not contain the decimal expansion of any proper divisor of k.

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