A154770 -id:A154770 - cp's OEIS frontend

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

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).

A225580 The sum of all substrings of n (including n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 66, 68, 70, 72, 74, 76, 78, 80

Offset: 1

Christian N. K. Anderson and Kevin L. Schwartz, May 10 2013

This sequence differs from A071980 beginning with n = 1010, and differs formulaically beginning with n = 1000 (the first four digit number). Where A071980 is calculated as a + ab + abc + abcd + bcd + cd + d for four digit numbers abcd, this sequence also includes the term bc in the sum.
Limits: n <= a(n) < 1.73*n. Proof: a(n)/n will be maximized when substrings are as large as possible while n is as small as possible, or for numbers of the form 199999999... The sum of substrings of this number is < 222222... + < 1234567... or < 3456790123.../2000000000... or < 1.728396.
The number 111 is the smallest term that occurs twice in the sequence, when n = {96, 100}. The number 2254 is the smallest term that occurs three times in the sequence, when n = {1476, 1510, 2008}.

			For n=1980, a(n) = 1 + 9 + 8 + 0 + 19 + 98 + 80 + 198 + 980 + 1980 = 3373. Note that A071980(1980) = 3258, because it does not include 9, 8, 98 in the sum.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
Christian N. K. Anderson, Ulam spiral of all values of a(n)<10000, color-coded by the number of times they occur.

Cf. A071980, A154770.

f:= proc(n) local i,d,L;
  L:= convert(n,base,10);
  d:= nops(L);
  add(L[i]*(d-i+1)*(10^i - 1)/9, i=1..d);
end proc:
map(f, [$1..100]); # Robert Israel, May 15 2025

Table[s = IntegerDigits[n]; Total[Flatten[Table[FromDigits /@ Partition[s, i, 1], {i, Length[s]}]]], {n, 100}] (* T. D. Noe, May 13 2013 *)

def a(n):
    s = str(n)
    return sum(int(s[i:j]) for j in range(1, len(s)+1) for i in range(j))
# David Radcliffe, May 15 2025

sapply(1:100,function(n) {tot=0; s=as.character(n); len=nchar(s); for(i in 1:len) for(j in i:len) tot=tot+as.numeric(substr(s,i,j)); tot})

a(n) = A138953(n) + n. (Note the offset in A138953 is zero. - Zak Seidov, May 16 2013)

a(n) = 11*a(floor(n/10)) - 10*a(floor(n/100)) + (n mod 10) * A055642(n). - David Radcliffe, May 15 2025

Example corrected by Zak Seidov, May 16 2013

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

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

PARI

Python

Formula

A225580 The sum of all substrings of n (including n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

R

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula