1,2

Equivalent definition: position of least significant

decimal digit in 16^n equal to one of {1,2,4,8}, or zero if none of

these occur. Since this is the true meaning, the sequence is flagged

"base", even if the actual definition does not explicitely refer to a

base.

This is related to A071071, i.e. the conjecture by

J. Shallit that {1,2,4,8,65536} is a minimal set for powers of 2

written in base 10. This amounts to say that any power of 2 contains

either a digit among {1,2,4,8}, or 65536 as subword (obtained by

deleting zero or more digits of the number).

Any power 2^k ends in a digit among {1,2,4,8} except for k=4n, n>0,

in which case 2^k=16^n ends in 6.

The present sequence gives the position of the first digit (starting

with units) of these powers which is among {1,2,4,8}. It appears as if

16^4=65536 would be the only power of 2 which does not contain any of

these four digits. This would imply the conjecture (and be an even

stronger statement).

The sequence has a repeating pattern of length 25 of the form

( 1 2 A B C 1 2 2 D E 1 F 2 G H 1 2 J 2 2 1 K 2 L M )... where A-M

follow again a repeating pattern:

A=(3,p,q,3,3)... ; C=(3,3,r,s,3)...; D=(t,3,3,3,u)... etc.

J. Shallit, Minimal primes, J. Recreational Mathematics 30 (2) (1999-2000), 113-117. Preprint available at www.cs.uwaterloo.ca/~shallit/papers.html.

a(1)=1 since 16^1 has a 1 in position 1 (10^1).

a(2)=2 since 16^2=256 has a 2 in position 2 (10^2).

a(3)=3 since 16^3=4096 has a 4 in position 3 (10^3).

a(4)=0 since 16^4=65536 has no digit 1,2,4 or 8.

If we arrange the terms in a 25 column matrix, we can see the pattern:

[1 2 3 0 3 1 2 2 5 4 1 4 2 4 4 1 2 4 2 2 1 3 2 7 5]

[1 2 4 3 3 1 2 2 3 4 1 4 2 4 3 1 2 4 2 2 1 5 2 5 3]

[1 2 5 7 4 1 2 2 3 4 1 3 2 7 3 1 2 4 2 2 1 12 2 3 7]

[1 2 3 5 7 1 2 2 3 4 1 3 2 5 3 1 2 3 2 2 1 6 2 3 5]

[1 2 3 4 3 1 2 2 5 3 1 3 2 3 8 1 2 5 2 2 1 4 2 3 4]

[1 2 3 4 3 1 2 2 6 5 1 6 2 5 4 1 2 5 2 2 1 3 2 5 4]

[1 2 7 3 3 1 2 2 3 7 1 8 2 7 3 1 2 5 2 2 1 7 2 4 3]

[1 2 4 5 4 1 2 2 3 7 1 3 2 4 3 1 2 4 2 2 1 4 2 3 5]

[1 2 3 4 4 1 2 2 3 7 1 3 2 6 3 1 2 3 2 2 1 4 2 3 8]

[1 2 3 5 3 1 2 2 4 3 1 3 2 3 4 1 2 6 2 2 1 5 2 3 6]

[1 2 3 4 3 1 2 2 9 4 1 5 2 4 5 1 2 8 2 2 1 3 2 4 5] etc.

(PARI) A137998(n, S=Vec("1248"))={ forstep( i=#n=Vec(Str(16^n)), 1, -1, setsearch(S, n[i]) & return( #n-i ))}

(PARI) t=0; matrix(25, 20/*# of rows*/, i, j, A137998(t++))~

Cf. A071071.

Sequence in context: A132385 A089235 A051910 this_sequence A080593 A051933 A131900

Adjacent sequences: A137995 A137996 A137997 this_sequence A137999 A138000 A138001

base,easy,nonn

M. F. Hasler (www.univ-ag.fr/~mhasler), Mar 26 2008