0,2

A "base" sequence visiting all the bases but nevertheless written here in base 10.

The number a(n) do not become its value in the new base, but becomes the base itself. So each term has a double status according to its preceding or following neighbor: regarding a(n-1), a(n) is a *base* (the least one not used so far) in which it is possible to read a(n-1); and regarding a(n+1), a(n) is a *number* to be read in the base expressed by a(n+1).

The first break, specific of this sequence written in base 10, occurs after a(9)=10. If, following the same principle, one build another sequence written, say in base 8, the beginning would be: 0,2,3,4,5,6,7,10,2,4... the first break occurring after a(7) instead of a(9). The inclusion of the unary base would lead to a different sequence since after the first occurrence of 11 would come 1 and not 2.

The word "walking base" refers to the "walking bass", a certain style of accompaniment in baroque music or jazz bass playing, in which the player, using a bass line composed of nonsyncopated notes of equal value, moves in stepwise motion to successive chord roots or notes, sometimes using passing notes.

The number base calculator,

Examples: The beginning is 0,2,3 but could also be 1,2,3.

a(0)=0. Now the least base in which 0 has a meaning is the binary base, so next term, a(1)=2.

The least base in which 2 makes sense is 3, so next term, a(2)=3.

The least base in which "10" makes sense is not base 11 but base 2, so next term, a(10)=2 (although 2 was used to read 0, it has not yet been used to read "10").

The least base in which this second 2 makes sense now is not 3 (because 3 has already been used to read a(1)=2), but 4, so next term a(11)=4.

a(101)=10: the least base not used so far to read "10" is base 10, so a(102)=10; then a(103)=11 (and although the value a(102)="10" in base 11 should be written "A", which is impossible in OEIS, this does not affect the next term a(103); anyway, this walking base is written all along in base 10, so a(102)=10).

Sequence in context: A093882 A138953 A053392 this_sequence A043270 A089898 A071785

Adjacent sequences: A114922 A114923 A114924 this_sequence A114926 A114927 A114928

base,nonn

Alexandre Wajnberg (alexandre.wajnberg(AT)skynet.be), Feb 20 2006