A342145 - OEIS

%I #27 Oct 25 2021 11:05:33

%S 0,9,10,11,12,15,19,20,23,24,25,29,30,31,36,37,39,40,48,49,50,51,52,

%T 57,59,60,61,62,69,70,71,73,74,75,78,79,80,81,86,87,89,90,91,94,95,96,

%U 97,98,99,100,101,102,103,104,105,106,107,108,109,110

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

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

%C 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.

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

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

%H Robert Israel, <a href="/A342145/b342145.txt">Table of n, a(n) for n = 1..10000</a>

%e 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.

%e 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.

%e See A342146 and A342147 for more examples.

%p 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:

%p select(filter, [$1..200]); # _Robert Israel_, Oct 24 2021

%o (PARI) select( is_A342135 = A341755, [0..199])

%o (Python)

%o def ok(n): return set(str(2*n+1)) & set(str(n)+str(n+1)) != set()

%o print([k for k in range(111) if ok(k)]) # _Michael S. Branicky_, Oct 24 2021

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

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

%K nonn,base

%O 1,2

%A _M. F. Hasler_, Mar 01 2021