A154771 -id:A154771 - cp's OEIS frontend

A154770 a(n+1) = A154771(a(n)) = sum of all distinct "valid substrings" of a(n); a(1)=10 (least nontrivial choice).

10, 11, 12, 15, 21, 24, 30, 33, 36, 45, 54, 63, 72, 81, 90, 99, 108, 127, 176, 283, 407, 458, 578, 733, 849, 1003, 1117, 1381, 2044, 2318, 2953, 4397, 5435, 6557, 7964, 9989, 12279, 16572, 26426, 36970, 49970, 67738, 84716, 100181, 111599, 139413, 209606

Offset: 1

Eric Angelini and M. F. Hasler, Jan 16 2009

"Valid substrings" means all numbers that appear as substring of n (written in decimal system). Starting values < 10 would yield a constant sequence.

			a(1) = 10 has { 0, 1, 10 } as distinct substrings,
a(2) = 0+1+10 = 11 has { 1, 11 } as distinct substrings,
a(3) = 1+11 = 12 has { 1, 2, 12 } as distinct substrings,
a(4) = 1+2+12 = 15 has { 1, 5, 15 } as distinct substrings.

E. Angelini, Sum of all valid substrings (VS), SeqFan list, Jan 16 2009

Cf. A004207, A154771.

A154770( n=2, a=10 ) = { local(d); while( n--, a=vecsort(concat(vector(d=#Str(a),i,vector(d,j,a%10^j)+(d--&!a\=10))),NULL,8);a*=vector(#a,i,1)~); a }
print1(a=10);for(i=1,50,print1("," a=A154770(,a)))

The word "distinct" added to definition Jan 19 2009 at the suggestion of Hugo van der Sanden.

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

Original entry on oeis.org

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

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

A154781 Sum of all numbers < n that appear as substring of n, written in decimal system.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 2, 5, 6, 7, 8, 9, 10, 11, 3, 4, 5, 3, 7, 8, 9, 10, 11, 12, 4, 5, 6, 7, 4, 9, 10, 11, 12, 13, 5, 6, 7, 8, 9, 5, 11, 12, 13, 14, 6, 7, 8, 9, 10, 11, 6, 13, 14, 15, 7, 8, 9, 10, 11, 12, 13, 7, 15, 16, 8, 9, 10, 11, 12, 13, 14, 15, 8, 17

Offset: 0

M. F. Hasler, Jan 16 2009

The condition "< n" narrows the meaning of "substring" to the strict sense, i.e., excludes n itself.

			Since n=0,...,9 has a single digit, only n itself appears as substring in n, thus a(n)=0.
10 has { 0, 1, 10 } as substrings, thus a(10) = 0+1 = 1.
11 has { 1, 11 } as substrings, thus a(11) = 1.
12 has { 1, 2, 12 } as substrings, thus a(12) = 1+2 = 3.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A154770, A154771.

san[n_]:=Total[Union[FromDigits/@Flatten[Table[Partition[IntegerDigits[n],i,1],{i,IntegerLength[n]-1}],1]]]; Array[san,90,0] (* Harvey P. Dale, May 27 2017 *)

A154781(n) = { local(d=#Str(n)); n=vecsort(concat(vector(d,i,vector(d,j,n%10^j)+(d--&!n\=10))),,12);n*vector(#n,i,i>1)~ }

def a(n):
    s = str(n)
    return sum(set(int(s[i:j]) for i in range(len(s)) for j in range(i+1, len(s)+1))) - n
print([a(n) for n in range(90)]) # Michael S. Branicky, Nov 08 2022

a(n) = A154771(n)-n. a(10^n) = A002275(n). a(n)>0 <=> n>9.

cp's OEIS Frontend

A154770 a(n+1) = A154771(a(n)) = sum of all distinct "valid substrings" of a(n); a(1)=10 (least nontrivial choice).

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

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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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Haskell

A154781 Sum of all numbers < n that appear as substring of n, written in decimal system.

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Examples

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Programs

Mathematica

PARI

Python

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