A120004 -id:A120004 - cp's OEIS frontend

A119999 Number of partitions of n into parts that occur in decimal representation as substrings of n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 8, 6, 5, 5, 4, 4, 4, 4, 2, 12, 2, 5, 8, 4, 6, 3, 5, 3, 2, 12, 7, 2, 4, 4, 8, 3, 3, 6, 2, 12, 12, 5, 2, 4, 5, 3, 8, 3, 2, 12, 7, 5, 4, 2, 3, 3, 3, 3, 2, 12, 12, 12, 7, 4, 2, 3, 4, 5, 2, 12, 7, 5, 4, 4, 3, 2, 3, 3, 2, 12, 12, 5, 12, 4, 5, 3, 2, 3, 2, 12, 7, 12, 4, 4, 7, 3

Offset: 0

Reinhard Zumkeller, Jun 13 2006

A120002 = first differences; A120003 = partial sums;

see A120000 and A120001 for records and where they occur: A120000(n)=a(A120001(n)).

			a(98) = #{98, 10*9+8, 2*9+10*8} = 3;
a(99) = #{99, 11*9} = 2;
a(100) = #{100, 10*10, 9*10+10*1, 8*10+20*1, 7*10+30*1, 6*10+40*1, 5*10+50*1, 4*10+60*1, 3*10+70*1, 2*10+80*1, 10+90*1, 100*1} = 12;
a(101) = #{101, 10*10+1, 9*10+11*1, 8*10+21*1, 7*10+31*1, 6*10+41*1, 5*10+51*1, 4*10+61*1, 3*10+71*1, 2*10+81*1, 10+91*1, 101*1} = 12;
a(102) = #{102, 10*10+2, 10*10+2*1, 9*10+6*2, ...} = 298.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A000041, A061827, A120004.

import Data.List (isInfixOf)
a119999 n = p (filter ((`isInfixOf` show n) . show) [1..n]) n where
   p _  0 = 1
   p [] _ = 0
   p ks'@(k:ks) m | m < k     = 0
                  | otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 14 2011

A218978 Table read by rows: n-th row lists all distinct substrings of decimal representation of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 10, 1, 11, 1, 2, 12, 1, 3, 13, 1, 4, 14, 1, 5, 15, 1, 6, 16, 1, 7, 17, 1, 8, 18, 1, 9, 19, 0, 2, 20, 1, 2, 21, 2, 22, 2, 3, 23, 2, 4, 24, 2, 5, 25, 2, 6, 26, 2, 7, 27, 2, 8, 28, 2, 9, 29, 0, 3, 30, 1, 3, 31, 2, 3, 32, 3

Offset: 0

Reinhard Zumkeller, May 02 2015, Nov 10 2012

A120004(n) = length of n-th row;

A154771(n) = sum of n-th row.

			Rows 100 .. 112:
.  100:  {0, 1, 10, 100},
.  101:  {0, 1, 10, 101},
.  102:  {0, 1, 2, 10, 102},
.  103:  {0, 1, 3, 10, 103},
.  104:  {0, 1, 4, 10, 104},
.  105:  {0, 1, 5, 10, 105},
.  106:  {0, 1, 6, 10, 106},
.  107:  {0, 1, 7, 10, 107},
.  108:  {0, 1, 8, 10, 108},
.  109:  {0, 1, 9, 10, 109},
.  110:  {0, 1, 10 ,11, 110},
.  111:  {1, 11, 111},
.  112:  {1, 2, 11, 12, 112}.

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

Cf. A031298, A219031 (squares in row), A262188 (palindromes in row).

import Data.List (inits, tails, sort, nub, genericIndex)
a218978 n k = a218978_row n !! k
a218978_row n = genericIndex a218978_tabf n
a218978_tabf = map (sort . nub . map (foldr (\d v -> 10 * v + d) 0) .
                   concatMap (tail . inits) . tails) a031298_tabf
-- Reinhard Zumkeller, corrected: Sep 15 2015, May 02 2015, Nov 10 2012

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).

A219032 Number of distinct squares as subwords of decimal representation of n-th square.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 2, 3, 2, 3, 4, 3, 2, 2, 2, 2, 3, 3, 3, 2, 2, 1, 2, 1, 2, 2, 3, 3, 3, 4, 4, 2, 4, 3, 4, 4, 2, 4, 4, 4, 5, 4, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 5, 4, 4, 3, 2, 2, 2, 4, 4, 2, 3, 2, 3, 6, 4, 3, 2, 2, 3, 1, 2, 3, 3, 5, 2, 2, 2, 2, 3

Offset: 0

Reinhard Zumkeller, Nov 10 2012

a(n) is the number of squares in n-th row of triangle A219031.

			.   n   row n in A219031
.  -----------------------------
.  20   [0, 4, 40, 400]            a(20) = #{0, 4, 400} = 3;
.  21   [1, 4, 41, 44, 441]        a(21) = #{1, 4, 441} = 3;
.  22   [4, 8, 48, 84, 484]        a(22) = #{4, 484} = 2;
.  23   [2, 5, 9, 29, 52, 529]     a(23) = #{9, 529} = 2;
.  24   [5, 6, 7, 57, 76, 576]     a(24) = #{576} = 1;
.  25   [2, 5, 6, 25, 62, 625]     a(25) = #{25, 625} = 2.

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

Cf. A010052, A120004, A219031.

a219032 = sum . map a010052 . a219031_row

from sympy import integer_nthroot
def A219032(n):
    s = str(n*n)
    m = len(s)
    return len(set(filter(lambda x: integer_nthroot(x,2)[1], (int(s[i:j]) for i in range(m) for j in range(i+1,m+1))))) # Chai Wah Wu, Oct 19 2021

a(n) = Sum_{k=0..A120004(n^2)} A010052(A219031(n,k)).

A079790 a(n) = number of m <= n which can be obtained by deleting digits from n.

Original entry on oeis.org

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, 5, 6, 6, 6, 6

Offset: 1

Amarnath Murthy, Feb 04 2003

a(n) = A120004(n) for n <= 100. a(101) = #{0, 1, 10, 11, 101} = 5; a(102) = #{0, 1, 2, 10, 12, 102} = 6. - Reinhard Zumkeller, Jun 15 2006

			a(124)=7 and the numbers are 1,2,4,12,14,24,124.

Robert Israel, Table of n, a(n) for n = 1..10000

f:= proc(n) local L,V;
  L:= convert(n,base,10);
  V:= convert(map(t -> L[t], combinat:-powerset([$1..nops(L)])),set) minus {[]};
  V:= map(proc(t) local i; add(t[i]*10^(i-1),i=1..nops(t)) end proc,V);
  nops(select(t -> t <= n, V))
end proc:
map(f, [$1..100]); # Robert Israel, Dec 03 2024

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 19 2003

Corrected and extended by Reinhard Zumkeller, Jun 15 2006

A120005 Smallest number containing exactly n distinct numbers in its decimal representation.

Original entry on oeis.org

0, 11, 10, 100, 102, 120, 1012, 1023, 1120, 1230, 10123, 10234, 11203, 11230, 12340, 101234, 102345, 111230, 112130, 112340, 123450, 1012345, 1023456, 1112130, 1112340, 1121340, 1123450, 1234560

Offset: 1

Reinhard Zumkeller, Jun 15 2006

A000030(a(n)) = 1 for n > 1;

A120004(a(n)) = n and A120004(m) < n for m < a(n).

			Illustration of initial values:
.   n |   a(n) ... and n-1 proper substrings
. ----+----------------------------------------------------
.   1 |     0
.   2 |    11    1
.   3 |    10    1    0
.   4 |   100   10    1   0
.   5 |   102   10    2   1   0
.   6 |   120   20   12   2   1   0
.   7 |  1012  101   12  10   2   1  0
.   8 |  1023  102   23  10   3   2  1  0
.   9 |  1120  120  112  20  12  11  2  1  0
.  10 |  1230  230  123  30  23  12  3  2  1  0
.  11 | 10123 1012  123 101  23  12 10  3  2  1 0
.  12 | 10234 1023  234 102  34  23 10  4  3  2 1 0
.  13 | 11203 1203 1120 203 120 112 20 12 11  3 2 1 0
.  14 | 11230 1230 1123 230 123 112 30 23 12 11 3 2 1 0
.  15 | 12340 2340 1234 340 234 123 40 34 23 12 4 3 2 1 0 .

import Data.List (isInfixOf, elemIndex)
import Data.Maybe (fromJust)
a120005 = fromJust . (`elemIndex` a120004_list)
-- Reinhard Zumkeller, Jul 16 2012

a(1) changed from 1 to 0 by Zak Seidov, May 30 2010

A348467 The number of distinct decimal representations of integers embedded as slices in the decimal representation of n!.

Original entry on oeis.org

1, 1, 1, 1, 3, 6, 6, 7, 11, 20, 25, 33, 32, 41, 60, 72, 80, 106, 104, 132, 140, 150, 173, 239, 241, 269, 306, 344, 369, 440, 487, 542, 550, 639, 639, 754, 799, 840, 777, 932, 1094, 1032, 1129, 1203, 1376, 1440, 1386, 1681, 1700, 1737, 1700, 1948, 1964, 2099, 2219

Offset: 0

Peter Luschny, Oct 19 2021

			0:  1 // 1;
1:  1 // 1;
2:  1 // 2;
3:  1 // 6;
4:  3 // 2,4,24;
5:  6 // 0,1,2,12,20,120;
6:  6 // 0,2,7,20,72,720;
7:  7 // 0,4,5,40,50,504,5040;
8: 11 // 0,2,3,4,20,32,40,320,403,4032,40320;
9: 20 // 0,2,3,6,8,28,36,62,80,88,288,362,628,880,2880,3628,6288,36288,62880, 362880.

Cf. A120004, A219032, A000142.

a[n_] := Length@ DeleteDuplicates[FromDigits /@ Rest@ Subsequences[ IntegerDigits[n!]]]; Array[a, 50, 0] (* Amiram Eldar, Oct 19 2021 *)

f(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); \\ A120004
a(n) = f(n!); \\ Michel Marcus, Oct 19 2021

from math import factorial
def A348467(n):
    s = str(factorial(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

a(n) = A120004(n!). - Michel Marcus, Oct 19 2021

cp's OEIS Frontend

A119999 Number of partitions of n into parts that occur in decimal representation as substrings of n.

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A218978 Table read by rows: n-th row lists all distinct substrings of 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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A219032 Number of distinct squares as subwords of decimal representation of n-th square.

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A079790 a(n) = number of m <= n which can be obtained by deleting digits from n.

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A120005 Smallest number containing exactly n distinct numbers in its decimal representation.

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Haskell

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A348467 The number of distinct decimal representations of integers embedded as slices in the decimal representation of n!.

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