A343951 Numbers with decimal expansion (d_1, ..., d_k) such that all the sums d_i + ... + d_j with 1 <= i <= j <= k are distinct.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 78, 79, 81, 82

Offset: 1

Author

Rémy Sigrist, May 05 2021

Keywords

• nonn
• base
• fini
• full

Comments

This sequence is finite, the last term being a(5562) = 8657913.

All positive terms are zeroless (A052382) and have distinct decimal digits (A010784).

There are 10, 72, 440, 1622, 2502, 906, 10, and 0 terms with 1..8 digits, resp. - Michael S. Branicky, May 05 2021

Examples

			Regarding 12458:
- we have the following partial sums of digits:
     i\j|  1  2  3  4  5
     ---+---------------
       1|  1  3  7 12 20
       2|  .  2  6 11 19
       3|  .  .  4  9 17
       4|  .  .  .  5 13
       5|  .  .  .  .  8
- as they are all distinct, 12458 is a term.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..5562

Crossrefs

Cf. A010784, A052382, A101274.

Programs

PARI

is(n) = { my (d=digits(n), s=setbinop((i,j)->vecsum(d[i..j]), [1..#d])); #s==#d*(#d+1)/2 }

Python

def ok(n):
  d, sums = str(n), set()
  for i in range(len(d)):
    for j in range(i, len(d)):
      sij = sum(map(int, d[i:j+1]))
      if sij in sums: return False
      else: sums.add(sij)
  return True
print(list(filter(ok, range(83)))) # Michael S. Branicky, May 05 2021