A236402 Numbers with property that the sum of any pair of adjacent digits is a substring of the number.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 400, 401, 402, 403, 404, 405, 406, 407

Offset: 1

Author

Eric Angelini, Jan 30 2014

Keywords

• nonn
• base
• easy

Comments

This sequence has density 1, since all numbers except a thin fraction have digits 0 through 18 in base 100. In particular, there are at most x^0.99782 non-members up to x for large enough x. (This can be improved.) - Charles R Greathouse IV, Jan 30 2014

1263907548 is the smallest term that contains all ten digits. - M. F. Hasler, Jan 30 2014

Where does this first differ from A032945? - R. J. Mathar, Feb 03 2014

This first differs from A032945 at a(110)=910 (followed by 1000, 1001, 1002, ...) while A032945(110)=1000 (followed by 1010, 1020, 1030, ...). - M. F. Hasler, Dec 28 2014

Examples

			Examples of numbers in the sequence:
80 --> 8+0=8
107 --> 1+0=1  /  0+7=7
910 --> 9+1=10  /  1+0=1
1037 --> 1+0=1  /  0+3=3  /  3+7=10
1459 --> 1+4=5  /  4+5=9  /  5+9=14
41358 --> 4+1=5  /  1+3=4  /  3+5=8  /  5+8=13
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..8495 (terms < 10^6)

Crossrefs

Cf. A236403 (complement).

Cf. A198298, A203565, A203566, A203569.

Programs

Mathematica

fQ[n_] := Module[{d, p, s}, d = IntegerDigits[n]; p = Partition[d, 2, 1]; s = Plus @@@ p; Complement[s, Union[d, FromDigits /@ p]] == {}]; Join[Range[0, 9], Select[Range[10, 1000], fQ]] (* T. D. Noe, Jan 30 2014 *)

PARI

is(n)=my(d=digits(n),S=Set(d),v=List(),t); for(i=2,#d, listput(v, 10*d[i-1]+d[i])); S=Set(concat(S,Vec(v))); for(i=2,#d, t=d[i-1]+d[i]; if(!setsearch(S, t), return(0))); 1 \\ Charles R Greathouse IV, Jan 13 2015

Python

def ok(n):
  s = str(n)
  return all(str(sum(map(int, s[i:i+2]))) in s for i in range(len(s)-1))
print(list(filter(ok, range(408)))) # Michael S. Branicky, Jun 11 2021

Formula

a(n) ~ n. - Charles R Greathouse IV, Jan 30 2014