A218978 -id:A218978 - cp's OEIS frontend

A120004 Number of distinct numbers as substrings of n in decimal representation.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 4, 4, 5, 5, 5, 5

Offset: 0

Reinhard Zumkeller, Jun 15 2006

a(n) = A079790(n) for n <= 100;
a(A120005(n)) = n and a(m) < n for m < A120005(n).
a(n) = length of n-th row in A218978; see also A154771. - Reinhard Zumkeller, Nov 10 2012

			n=101: {'1','0','10','01','101'} -> a(101) = #{0,1,10,101} = 4;
n=102: {'1','0','2','10','02','102'} -> a(102) = #{0,1,2,10,102} = 5.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A119999.

import Data.List (isInfixOf)
a120004 n = sum $ map (fromEnum . (`isInfixOf` show n) . show) [0..n]
-- Reinhard Zumkeller, Jul 16 2012, Aug 14 2011

a[n_] := Length@ DeleteDuplicates[FromDigits /@ Rest@ Subsequences[ IntegerDigits[n]]]; Array[a, 100, 0] (* Amiram Eldar, Oct 19 2021 *)

a(n) = if (n==0, return (1)); my(d=digits(n), list=List()); for (k=1, #d, for (j=1, #d-k+1, my(dk=vector(j, i, d[k+i-1])); listput(list, fromdigits(dk)););); #Set(list); \\

def A120004(n):
    s = str(n)
    m = len(s)
    return len(set(int(s[i:j]) for i in range(m) for j in range(i+1,m+1))) # Chai Wah Wu, Oct 19 2021

Offset changed and a(0)=1 inserted, in consequence of Zak Seidov's correction in A120005. - Reinhard Zumkeller, May 30 2010

A262188 Table read by rows: row n contains all distinct palindromes contained as substrings in decimal representation of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 1, 11, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 0, 2, 1, 2, 2, 22, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 0, 3, 1, 3, 2, 3, 3, 33, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 0, 4, 1, 4, 2, 4, 3, 4, 4, 44, 4, 5, 4, 6, 4

Offset: 0

Reinhard Zumkeller, Sep 14 2015

Length of row n = A262190(n);
T(n,0) = A054054(n);
T(n,A262190(n)-1) = A047813(n).

			.     n |  T(n,*)           n |  T(n,*)              n |  T(n,*)
.  -----+-----------    ------+-------------    -------+--------------
.   100 |  0,1           1000 |  0,1             10000 |  0,1
.   101 |  0,1,101       1001 |  0,1,1001        10001 |  0,1,10001
.   102 |  0,1,2         1002 |  0,1,2           10002 |  0,1,2
.   103 |  0,1,3         1003 |  0,1,3           10003 |  0,1,3
.   104 |  0,1,4         1004 |  0,1,4           10004 |  0,1,4
.   105 |  0,1,5         1005 |  0,1,5           10005 |  0,1,5
.   106 |  0,1,6         1006 |  0,1,6           10006 |  0,1,6
.   107 |  0,1,7         1007 |  0,1,7           10007 |  0,1,7
.   108 |  0,1,8         1008 |  0,1,8           10008 |  0,1,8
.   109 |  0,1,9         1009 |  0,1,9           10009 |  0,1,9
.   110 |  0,1,11        1010 |  0,1,101         10010 |  0,1,1001
.   111 |  1,11,111      1011 |  0,1,11,101      10011 |  0,1,11,1001
.   112 |  1,2,11        1012 |  0,1,2,101       10012 |  0,1,2,1001
.   113 |  1,3,11        1013 |  0,1,3,101       10013 |  0,1,3,1001
.   114 |  1,4,11        1014 |  0,1,4,101       10014 |  0,1,4,1001
.   115 |  1,5,11        1015 |  0,1,5,101       10015 |  0,1,5,1001
.   116 |  1,6,11        1016 |  0,1,6,101       10016 |  0,1,6,1001
.   117 |  1,7,11        1017 |  0,1,7,101       10017 |  0,1,7,1001
.   118 |  1,8,11        1018 |  0,1,8,101       10018 |  0,1,8,1001
.   119 |  1,9,11        1019 |  0,1,9,101       10019 |  0,1,9,1001
.   120 |  0,1,2         1020 |  0,1,2           10020 |  0,1,2
.   121 |  1,2,121       1021 |  0,1,2           10021 |  0,1,2
.   122 |  1,2,22        1022 |  0,1,2,22        10022 |  0,1,2,22
.   123 |  1,2,3         1023 |  0,1,2,3         10023 |  0,1,2,3
.   124 |  1,2,4         1024 |  0,1,2,4         10024 |  0,1,2,4
.   125 |  1,2,5         1025 |  0,1,2,5         10025 |  0,1,2,5  .

Reinhard Zumkeller, Rows n = 0..10000 of triangle, flattened
Index entries for sequences related to palindromes

Cf. A262190 (row lengths), A054054 (left edge), A047813 (right edge), A136522, A002113.

Cf. A031298, A218978.

import Data.List (inits, tails, nub, sort)
a262188 n k = a262188_tabf !! n !! k
a262188_row n = a262188_tabf !! n
a262188_tabf = map (sort . nub . map (foldr (\d v -> 10 * v + d) 0) .
   filter (\xs -> length xs == 1 || last xs > 0 && reverse xs == xs) .
          concatMap (tail . inits) . tails) a031298_tabf

A262188_row(n,b=10)=Set(concat(vector(logint(n+!n,b)+1,m,m=n\=b^(m>1);select(is_A002113,vector(logint(m+!m,b)+1,k,m%b^k))))) \\ M. F. Hasler, Jun 19 2018

A154771 Sum of all numbers that appear as substring of n, written in decimal system.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 17, 19, 21, 23, 25, 27, 29, 22, 24, 24, 28, 30, 32, 34, 36, 38, 40, 33, 35, 37, 36, 41, 43, 45, 47, 49, 51, 44, 46, 48, 50, 48, 54, 56, 58, 60, 62, 55, 57, 59, 61, 63, 60, 67, 69, 71, 73, 66, 68, 70, 72, 74, 76, 72, 80, 82, 84, 77, 79, 81

Offset: 0

M. F. Hasler, Jan 16 2009

a(n) is the sum of n-th row in A218978; see also A120004. - Reinhard Zumkeller, Nov 10 2012

			Since n=0,...,9 has a single digit, only n itself appears as substring in k, thus a(n)=n.
10 has { 0, 1, 10 } as substrings, thus a(10) = 0+1+10 = 11.
11 has { 1, 11 } as substrings, thus a(11) = 1+11 = 12.
12 has { 1, 2, 12 } as substrings, thus a(12) = 1+2+12 = 15.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A154770, A154781.

a154771 = sum . a218978_row :: Integer -> Integer
-- Reinhard Zumkeller, Nov 10 2012

A154771(n) = { local(d=#Str(n)); n=vecsort(concat(vector(d,i,vector(d,j,n%10^j)+(d--&!n\=10))),,8);n*vector(#n,i,1)~ }

def a(n):
    s = str(n); L = len(s)
    return sum(set(int(s[i:j]) for i in range(L) for j in range(i+1, L+1)))
print([a(n) for n in range(73)]) # Michael S. Branicky, Nov 08 2022

a(n) = n+A154781(n).

a(10^n) = A002275(n+1).

A265183 Base-10 analog of Marko Riedel's A265008, but allowing A, B, C to be zero.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 8, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 0, 0, 1, 1, 0, 0, 0, 0, 5, 5, 0, 0, 0, 0, 1, 0, 0, 1, 5, 5, 1, 0, 0, 0, 0, 0, 0, 0, 5, 5, 0, 0, 0, 0, 0, 0, 0, 0, 5, 5, 0, 0, 0, 0, 0, 0, 0, 0, 5, 5, 0, 0, 0, 0, 0, 0, 0, 0, 5, 5, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Dec 04 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A265008, A265182, A265236.

Cf. A218978.

a265183 n = length [() | let cs = a218978_row n, a <- cs, b <- cs, c <- cs,
                         a * b == c || c == 0 && a * b == 0]
-- Reinhard Zumkeller, Dec 05 2015

Data corrected by Reinhard Zumkeller, Dec 05 2015

A219031 Table read by rows: n-th row lists all distinct subwords of decimal representation of n-th square.

Original entry on oeis.org

0, 1, 4, 9, 1, 6, 16, 2, 5, 25, 3, 6, 36, 4, 9, 49, 4, 6, 64, 1, 8, 81, 0, 1, 10, 100, 1, 2, 12, 21, 121, 1, 4, 14, 44, 144, 1, 6, 9, 16, 69, 169, 1, 6, 9, 19, 96, 196, 2, 5, 22, 25, 225, 2, 5, 6, 25, 56, 256, 2, 8, 9, 28, 89, 289, 2, 3, 4, 24, 32, 324, 1, 3

Offset: 0

Reinhard Zumkeller, Nov 10 2012

A219032(n) gives number of squares in n-th row.

			First 50 rows of triangle:
.   0: [0]                      .  25: [2,5,6,25,62,625]
.   1: [1]                      .  26: [6,7,67,76,676]
.   2: [4]                      .  27: [2,7,9,29,72,729]
.   3: [9]                      .  28: [4,7,8,78,84,784]
.   4: [1,6,16]                 .  29: [1,4,8,41,84,841]
.   5: [2,5,25]                 .  30: [0,9,90,900]
.   6: [3,6,36]                 .  31: [1,6,9,61,96,961]
.   7: [4,9,49]                 .  32: [0,1,2,4,10,24,102,1024]
.   8: [4,6,64]                 .  33: [0,1,8,9,10,89,108,1089]
.   9: [1,8,81]                 .  34: [1,5,6,11,15,56,115,156,1156]
.  10: [0,1,10,100]             .  35: [1,2,5,12,22,25,122,225,1225]
.  11: [1,2,12,21,121]          .  36: [1,2,6,9,12,29,96,129,296,1296]
.  12: [1,4,14,44,144]          .  37: [1,3,6,9,13,36,69,136,369,1369]
.  13: [1,6,9,16,69,169]        .  38: [1,4,14,44,144,444,1444]
.  14: [1,6,9,19,96,196]        .  39: [1,2,5,15,21,52,152,521,1521]
.  15: [2,5,22,25,225]          .  40: [0,1,6,16,60,160,600,1600]
.  16: [2,5,6,25,56,256]        .  41: [1,6,8,16,68,81,168,681,1681]
.  17: [2,8,9,28,89,289]        .  42: [1,4,6,7,17,64,76,176,764,1764]
.  18: [2,3,4,24,32,324]        .  43: [1,4,8,9,18,49,84,184,849,1849]
.  19: [1,3,6,36,61,361]        .  44: [1,3,6,9,19,36,93,193,936,1936]
.  20: [0,4,40,400]             .  45: [0,2,5,20,25,202,2025]
.  21: [1,4,41,44,441]          .  46: [1,2,6,11,16,21,116,211,2116]
.  22: [4,8,48,84,484]          .  47: [0,2,9,20,22,209,220,2209]
.  23: [2,5,9,29,52,529]        .  48: [0,2,3,4,23,30,230,304,2304]
.  24: [5,6,7,57,76,576]        .  49: [0,1,2,4,24,40,240,401,2401] .

Reinhard Zumkeller, Rows n=0..1000 of triangle, flattened

Cf. A218978, A000290.

a219031 n k = a219031_tabf !! n !! k
a219031_row n = a219031_tabf !! n
a219031_tabf = map a218978_row a000290_list

A265182 Base-10 analog of Marko Riedel's A265008.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 0, 5, 0, 0, 1, 1, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 1, 0, 0, 1, 0, 5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0

Offset: 1

N. J. A. Sloane, Dec 04 2015

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

Cf. A265008, A265236.
See A265183 for the version where A, B, C may be zero.
Cf. A218978.

a265182 n = length [() | let cs = dropWhile (== 0) $ a218978_row n, c <- cs,
            let as = takeWhile (<= c) cs, a <- as, b <- as, a * b == c]
-- Reinhard Zumkeller, Dec 05 2015

F:= proc(n) local L, ss;
  L:= convert(n, base, 10);
  ss:= {seq(seq(add(10^(i-j)*L[i], i=j..k), j=1..k), k=1..nops(L))} minus {0};
  numboccur(true, [seq(seq(member(a*b, ss), a=ss), b=ss)]);
end proc:
seq(F(n), n=1..1000); # Robert Israel, Dec 06 2015

Corrected by Lars Blomberg, Dec 05 2015

A383592 Positive integers k divisible by all positive integers whose decimal expansion appears as a substring of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 20, 22, 24, 30, 33, 36, 40, 44, 48, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 110, 120, 150, 200, 210, 220, 240, 250, 300, 330, 360, 400, 420, 440, 480, 500, 510, 520, 550, 600, 630, 660, 700, 770, 800, 840, 880

Offset: 1

Rémy Sigrist, May 01 2025

This sequence is infinite as ten times a term is also a term.
All terms are of the form A037124(k) or A037124(k) + d where k > 0 and d divides A037124(k) while having strictly less decimal digits as A037124(k).
Empirically, all terms have either one or two nonzero decimal digits.

			The number 240 is divisible by 2, 24, 240, 4 and 40, so 240 belongs to this sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10314
Rémy Sigrist, PARI program

Cf. A037124, A078546, A175381 (binary variant), A178157, A218978.

Select[Range[880],AllTrue[#/Select[FromDigits/@Subsequences[IntegerDigits[#]],#>0&],IntegerQ]&] (* James C. McMahon, May 13 2025 *)

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

PARI
```
\\ See Links section.
```

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 all(n%v == 0 for ss in subs if (v:=int(ss)) > 0)
print([k for k in range(1000) if ok(k)]) # Michael S. Branicky, May 09 2025

cp's OEIS Frontend

A120004 Number of distinct numbers as substrings of n in decimal representation.

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A262188 Table read by rows: row n contains all distinct palindromes contained as substrings in decimal representation of n.

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A154771 Sum of all numbers that appear as substring of n, written in decimal system.

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A265183 Base-10 analog of Marko Riedel's A265008, but allowing A, B, C to be zero.

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A219031 Table read by rows: n-th row lists all distinct subwords of decimal representation of n-th square.

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A265182 Base-10 analog of Marko Riedel's A265008.

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Maple

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A383592 Positive integers k divisible by all positive integers whose decimal expansion appears as a substring of k.

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