A342145 Numbers k such that k + (k+1) = 2*k + 1 shares at least one digit with either k or k+1.

0, 9, 10, 11, 12, 15, 19, 20, 23, 24, 25, 29, 30, 31, 36, 37, 39, 40, 48, 49, 50, 51, 52, 57, 59, 60, 61, 62, 69, 70, 71, 73, 74, 75, 78, 79, 80, 81, 86, 87, 89, 90, 91, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110

Offset

1,2

Comments

The terms come frequently in runs of consecutive integers; sequence A341755 gives the length of the run + 1.

In particular, all intervals [10^m-1, ..., 10^m+(10^m-1)/9], m >= 1, e.g., {9, 11}, {99, ..., 111}, {999, ..., 1111}, ..., are subsequences.

See A342146 and A342147 for the variants where 2k+1 shares a digit with k, respectively with k+1. Their union equals this sequence.

Almost all numbers are in this sequence: It has asymptotic density 1, since almost all large enough numbers are pandigital.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

0 and 9 are in the sequence because 0 + 1 = 1 and 9 + 10 = 19 share a digit with {0, 1} and with {9, 10}, respectively.
1 and 8 and 13 are not in the sequence because 1 + 2 = 3, 8 + 9 = 17 and 13 + 14 = 27 do not share a digit with the respective right hand side.
See A342146 and A342147 for more examples.

Maple

filter:= proc(n) convert(convert(2*n+1, base, 10), set) intersect(convert(convert(n, base, 10), set) union convert(convert(n+1, base, 10), set)) <> {} end proc:
select(filter, [$1..200]); # Robert Israel, Oct 24 2021

Prog

(PARI) select( is_A342135 = A341755, [0..199])
(Python)
def ok(n): return set(str(2*n+1)) & set(str(n)+str(n+1)) != set()
print([k for k in range(111) if ok(k)]) # Michael S. Branicky, Oct 24 2021

Crossrefs

Cf. A341755, A341756, A341757; A342146 and A342147 (variants).

Cf. A002275 (repunits: (10^n-1)/9), A005408 (odd numbers: 2n+1).

Sequence in context: A258469 A360538 A173147 * A293761 A253881 A097480

Adjacent sequences: A342142 A342143 A342144 * A342146 A342147 A342148

Keyword

nonn,base

Author

M. F. Hasler, Mar 01 2021

Status

approved