A121041 -id:A121041 - cp's OEIS frontend

A293878 Numbers having '18' as substring of their digits / decimal expansion.

18, 118, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 218, 318, 418, 518, 618, 718, 818, 918, 1018, 1118, 1180, 1181, 1182, 1183, 1184, 1185, 1186, 1187, 1188, 1189, 1218, 1318, 1418, 1518, 1618, 1718, 1800, 1801, 1802, 1803, 1804, 1805, 1806, 1807, 1808, 1809, 1810, 1811, 1812

Offset: 1

M. F. Hasler, Oct 18 2017

Row 16 of A292690 and A293869. A121038 lists the terms which are divisible by 18.

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

Cf. A292690, A293869.
Cf. A011540, A011531, A011532, A011533, A011534, A011535, A011536, A011537, A011538, A011539: analog for '0' - '9'.
Cf. A293870, A293871, A293872, A293873, A293874, A293875, A293876, A293877, A293879, A293880: same for '10' - '20'.
Cf. A121041, A121022, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040: subsequences of the above, containing only multiples of the pattern p.

Select[Range[2000], StringContainsQ[IntegerString[#], "18"] &] (* Paolo Xausa, Feb 25 2024 *)

is_A293878 = has(n, p=18, m=10^#Str(p))=until(p>n\=10, n%m==p&&return(1))

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

A293879 Numbers having '19' as substring of their digits.

Original entry on oeis.org

19, 119, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 219, 319, 419, 519, 619, 719, 819, 919, 1019, 1119, 1190, 1191, 1192, 1193, 1194, 1195, 1196, 1197, 1198, 1199, 1219, 1319, 1419, 1519, 1619, 1719, 1819, 1900, 1901, 1902, 1903, 1904, 1905, 1906, 1907, 1908, 1909, 1910, 1911

Offset: 1

M. F. Hasler, Oct 18 2017

Row 19 of A292690 and A293869. A121039 lists the terms which are divisible by 19.

Index entries for 10-automatic sequences

Cf. A292690, A293869.
Cf. A011540, A011531, A011532, A011533, A011534, A011535, A011536, A011537, A011538, A011539: analog for '0' - '9'.
Cf. A293870, A293871, A293872, A293873, A293874, A293875, A293876, A293877, A293878, A293880: same for '10' - '20'.
Cf. A121041, A121022, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040: subsequences of the above, containing only multiples of the pattern p.

Select[Range[2000],SequenceCount[IntegerDigits[#],{1,9}]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 11 2019 *)

is_A293879 = has(n, p=19, m=10^#Str(p))=until(p>n\=10, n%m==p&&return(1))

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

A293873 Numbers having '13' as substring of their digits.

Original entry on oeis.org

13, 113, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 213, 313, 413, 513, 613, 713, 813, 913, 1013, 1113, 1130, 1131, 1132, 1133, 1134, 1135, 1136, 1137, 1138, 1139, 1213, 1300, 1301, 1302, 1303, 1304, 1305, 1306, 1307, 1308, 1309, 1310, 1311, 1312, 1313, 1314, 1315

Offset: 1

M. F. Hasler, Oct 18 2017

Row 13 of A292690 and A293869. A121033 is the subsequence of multiples of 13.

Index entries for 10-automatic sequences

Cf. A292690, A293869.
Cf. A011540, A011531, A011532, A011533, A011534, A011535, A011536, A011537, A011538, A011539: analog for '0' - '9'.
Cf. A293870, A293871, A293872, A293874, A293875, A293876, A293877, A293878, A293879, A293880: same for '10' - '20'.
Cf. A121041, A121022, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040: subsequences of the above, containing only multiples of the pattern p.

Select[Range[1350],SequenceCount[IntegerDigits[#],{1,3}]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 31 2017 *)

is_A293873 = has(n, p=13, m=10^#Str(p))=until(p>n\=10, n%m==p&&return(1))

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

A293880 Numbers having '20' as substring of their digits.

Original entry on oeis.org

20, 120, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 220, 320, 420, 520, 620, 720, 820, 920, 1020, 1120, 1200, 1201, 1202, 1203, 1204, 1205, 1206, 1207, 1208, 1209, 1220, 1320, 1420, 1520, 1620, 1720, 1820, 1920, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010

Offset: 1

M. F. Hasler, Oct 18 2017

Row 20 of A292690 and A293869. A121040 lists the terms which are divisible by 19.

Index entries for 10-automatic sequences

Cf. A292690, A293869.
Cf. A011540, A011531, A011532, A011533, A011534, A011535, A011536, A011537, A011538, A011539: analog for '0' - '9'.
Cf. A293870, A293871, A293872, A293873, A293874, A293875, A293876, A293877, A293878, A293879: same for '10' - '19'.
Cf. A121041, A121022, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040: subsequences of the above, containing only multiples of the pattern p.

Select[Range[2100],SequenceCount[IntegerDigits[#],{2,0}]>0&] (* Harvey P. Dale, Jul 25 2021 *)

is_A293880 = has(n, p=20, m=10^#Str(p))=until(p>n\=10, n%m==p&&return(1))

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

A121042 Smallest divisor of n that is also contained in the decimal representation of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 23, 2, 5, 2, 27, 2, 29, 3, 1, 2, 3, 34, 5, 3, 37, 38, 3, 4, 1, 2, 43, 4, 5, 46, 47, 4, 49, 5, 1, 2, 53, 54, 5, 56, 57, 58, 59, 6, 1, 2, 3, 4, 5, 6, 67, 68, 69, 7, 1, 2, 73, 74, 5, 76, 7, 78, 79, 8, 1, 2, 83, 4, 5, 86, 87, 8, 89, 9, 1

Offset: 1

Reinhard Zumkeller, Jul 21 2006

1 <= a(n) <= n;
a(A011531(n)) = 1; a(n) = n iff A121041(n) = 1.
a(n) = 1 for almost all n (measure 1). - Charles R Greathouse IV, Mar 31 2016

			a(48) = Min{4, 8, 48} = 4;
a(49) = Min{49} = 49;
a(120) = Min{1, 2, 12, 20, 120} = 1;
a(121) = Min{1} = 1.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A011531, A027750, A121041, A383749 (fixed points).

A121042[n_] := SelectFirst[Divisors[n], StringContainsQ[IntegerString[n], IntegerString[#]] &];
Array[A121042, 100] (* Paolo Xausa, May 12 2025 *)

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)=fordiv(n,d, if(substr(n,d), return(d))) \\ Charles R Greathouse IV, Mar 31 2016

A239058 Numbers whose divisors all appear as a substring in their decimal expansion.

Original entry on oeis.org

1, 11, 13, 17, 19, 31, 41, 61, 71, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 241, 251, 271, 281, 311, 313, 317, 331, 401, 419, 421, 431, 461, 491, 521, 541, 571, 601, 613, 617, 619, 631, 641, 661, 691, 701, 719, 751, 761, 811, 821, 881, 911, 919, 941, 971

Offset: 1

M. F. Hasler, Mar 09 2014

A subsequence of A092911 (all divisors can be formed using the digits of the number) which is a subsequence of A011531 (numbers having the digit 1).
Are 1 and 125 the only nonprime terms in this sequence?
No: 17692313, 4482669527413081, 21465097175420089, and 567533481816008761 are members. - Charles R Greathouse IV, Mar 09 2014
See A239060 for the nonprime terms of this sequence, which include in particular the squares of terms of A115738 (unless such a square would not have a digit 1).

			All primes having the digit 1 (A208270) are in this sequence, because {1, p} are the only divisors of a prime p.
The divisors of 125 are {1, 5, 25, 125}; it can be seen that all of them occur as a substring in 125, therefore 125 is in this sequence.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to decimal expansion of n

Cf. A092911, A011531, A121041, A121022-A121040, A018834.

is(n,d=vecextract(divisors(n),"^-1"))={ setminus(select(x->x<10,d),Set(digits(n)))&&return;!for(L=2,#Str(d[#d]),setminus(select(x->x
<10^L&&x>=10^(L-1),d),Set(concat(vector(L,o,digits(n\10^(L-o),10^L)))))&&return)}

overlap(long,short)=my(D=10^#digits(short)); while(long>=short, if(long%D==short,return(1));long\=10); 0
is(n)=my(d=divisors(n)); forstep(i=#d-1,1,-1, if(!overlap(n,d[i]), return(0))); 1 \\ Charles R Greathouse IV, Mar 09 2014

A239060 Nonprime numbers whose divisors all appear as a substring in the number's decimal expansion.

Original entry on oeis.org

1, 125, 17692313

Offset: 1

M. F. Hasler, Mar 09 2014

This is the subsequence of A239058 without the primes having a digit 1, A208270. It is thus a subsequence of A092911 (all divisors can be formed using the digits of the number) which is a subsequence of A011531 (numbers having the digit 1).
The term a(3)=17692313=A239058(870356), as well as the numbers 4482669527413081, 21465097175420089, and 567533481816008761 which are also members, were found by Charles R Greathouse IV, Mar 09 2014
The square of any term of A115738 is a member of this sequence. The above larger examples are of that form.
a(4) > 10^12. - Giovanni Resta, Sep 08 2018

			The divisors of 17692313 are {1, 23, 769231, 17692313}; it can be seen that all of them occur as a substring in 17692313, therefore 17692313 is in this sequence.

Cf. A092911, A011531, A121041, A121022-A121040, A018834.

PARI
```
is(n)=!isprime(n)&&is_A239058(n)
```

overlap(long,short)=my(D=10^#digits(short)); while(long>=short, if(long%D==short,return(1));long\=10); 0
is(n)=my(d=divisors(n)); #d!=2 && !forstep(i=#d-1,1,-1, if(!overlap(n,d[i]), return(0))) \\ Charles R Greathouse IV, Mar 09 2014

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.

A155005 Smallest number having exactly n divisors that are contained in its decimal representation.

Original entry on oeis.org

1, 10, 12, 110, 120, 1020, 1200, 1248, 10250, 11250, 12480, 31248, 132600, 124800, 112500, 312480, 1248000, 1312500, 1125000, 3124800, 14437500, 16250000, 11250000, 31248000, 103125000, 144375000, 112500000, 131250000

Offset: 1

Reinhard Zumkeller, Jan 18 2009

A121041(a(n)) = n and A121041(m) < n for m < a(n).

Conjecture: a(5+5n)=1125*10^n for n>0. [Robert G. Wilson v, Jan 23 2009]

			a(4) = 110, A121041(110) = #{1, 10, 11, 110} = 4;
a(5) = 120, A121041(120) = #{1, 2, 12, 20, 120} = 5;
a(6) = 1020, A121041(1020) = #{1, 2, 10, 20, 102, 1020} = 6.

f[n_] := Block[{d = Divisors@ n, len = DivisorSigma[0, n], i = 1, c = 1, s = ToString@ n}, While[i < len, If[ StringMatchQ[s, "*" <> ToString[d[[i]]] <> "*"], c++ ] ; i++ ]; c]; t = Table[0, {30}]; Do[ a = f[n]; If[ t[[a]] == 0, t[[a]] = n; Print[{a, n}]], {n, 2*10^8}] (* Robert G. Wilson v, Jan 23 2009 *)
dcdr[n_]:=Count[Divisors[n],?(SequenceCount[IntegerDigits[n],IntegerDigits[#]]>0&)]; DeleteDuplicates[Table[{n,dcdr[n]},{n,1313*10^5}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* _Harvey P. Dale, Feb 04 2025 *)

a(17)-a(28) from Robert G. Wilson v, Jan 23 2009

cp's OEIS Frontend

A293878 Numbers having '18' as substring of their digits / decimal expansion.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A293879 Numbers having '19' as substring of their digits.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A293873 Numbers having '13' as substring of their digits.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A293880 Numbers having '20' as substring of their digits.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A121042 Smallest divisor of n that is also contained in the decimal representation of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A239058 Numbers whose divisors all appear as a substring in their decimal expansion.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

A239060 Nonprime numbers whose divisors all appear as a substring in the number's decimal expansion.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

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

Views