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

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

0,13

Comments

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

Links

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

Formula

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

Example

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.

Mathematica

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 *)

Prog

(PARI) 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)~ }
(Python)
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

Crossrefs

Cf. A154770, A154771.

Sequence in context: A077396 A243196 A247485 * A118716 A004177 A004721

Adjacent sequences: A154778 A154779 A154780 * A154782 A154783 A154784

Keyword

base,easy,nonn

Author

M. F. Hasler, Jan 16 2009

Status

approved