A387069 Numbers whose representation in base b are their representation in base b+1 with a "0" added at the end for some b.

10, 12, 273, 546, 582, 11808655, 34503520, 35555155, 54813136653716, 1066338805156287287067, 1124161332414632881704, 2305867155177711644802, 2306166776784312535170, 5744341611556736174883, 2705154287309969771123575182312

Offset

1,1

Comments

The corresponding b's are: 2, 2, 3, 3, 3, 5, 5, 5, 7, 9, 9, 9, 9, 9, 11, ...

From Sean A. Irvine, Aug 18 2025: (Start)

Numbers k such that k = Sum_{j>=1} d(j) * b^j = Sum_{j>=0} d(j+1) * (b+1)^j where 0 <= d(j) < b.

For k to be a solution in base b, we require Sum d(j) = 0 (mod b) and Sum_{j>=2} (-1)^j*d(j) = 2*d(1) (mod b+1). (End)

Links

Table of n, a(n) for n=1..15.

Sean A. Irvine, Python program

Example

a(1) = 10, is 1010 in base 2 and 101 in base 3.
a(2) = 12, is 1100 in base 2 and 110 in base 3.
a(3) = 273, is 101010 in base 3 and 10101 in base 4.

Prog

(PARI) isok(k) = for (b=2, k-1, if (digits(k, b) == concat(digits(k, b+1), 0), return(b)); ); \\ Michel Marcus, Aug 16 2025

Crossrefs

Cf. A062845, A062853, A062928.

Sequence in context: A087392 A255534 A262422 * A248336 A332478 A144814

Adjacent sequences: A387066 A387067 A387068 * A387070 A387071 A387072

Keyword

nonn,base,more

Author

Cristiano Campos de Oliveira, Aug 15 2025

Extensions

a(9)-a(15) from Sean A. Irvine, Aug 18 2025

Status

approved