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

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

Author

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

Keywords

• nonn
• base

Comments

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}.

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A071980, A154770.

Programs

Maple

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

Mathematica

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

Python

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

R

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})

Formula

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

Extensions

Example corrected by Zak Seidov, May 16 2013