A352924 In the decimal expansion of 2^n, the proper substring which is the largest power of 2 is 2^a(n); a(n) = -1 if no proper substring of 2^n is a power of 2.

-1, -1, -1, -1, 0, 1, 2, 3, 1, 1, 2, 3, 2, 3, 4, 5, -1, 1, 2, 3, 3, 1, 2, 3, 4, 5, 6, 3, 3, 3, 3, 6, 2, 3, 3, 3, 3, 3, 3, 3, 4, 5, 3, 3, 4, 5, 6, 5, 3, 2, 3, 3, 2, 2, 3, 3, 2, 3, 3, 6, 3, 3, 4, 3, 4, 5, 6, 6, 3, 3, 4, 5, 6, 5, 3, 4, 5, 6, 2, 3, 3, 4, 5, 6, 4, 5

Offset

0,7

Links

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Example

The initial powers of 2, the largest proper substrings 2^k (or a dash "-"), and k (or -1):
   n       2^n    2^k    k
   0         1     -    -1
   1         2     -    -1
   2         4     -    -1
   3         8     -    -1
   4        16     1     0
   5        32     2     1
   6        64     4     2
   7       128     8     3
   8       256     2     1
   9       512     2     1
  10      1024     4     2
  11      2048     8     3
  12      4096     4     2
  13      8192     8     3
  14     16384    16     4
  15     32768    32     5
  16     65536     -    -1
  17    131072     2     1
  18    262144     4     2
  19    524288     8     3
  20   1048576     8     3
  ...

Mathematica

Table[If[Length[#] == 0, -1, #[[-1, -1]]] &@ Reap[Block[{w = IntegerString[2^n], k = 0}, While[k < n, If[StringContainsQ[w, IntegerString[2^k]], Sow[k]]; k++]]][[-1]], {n, 0, 85}] (* Michael De Vlieger, Apr 26 2022 *)

Prog

(PARI) a(n) = {my(s = Str(2^n)); forstep (k=n-1, 0, -1, if (#strsplit(s, Str(2^k)) >= 2, return(k)); ); return(-1); \\ Michel Marcus, Apr 27 2022
(Python)
def a(n):
    s = str(2**n)
    for k in range(n-1, -1, -1):
        if str(2**k) in s: return k
    return -1
print([a(n) for n in range(86)]) # Michael S. Branicky, Apr 27 2022

Crossrefs

Cf. A000079.

Sequence in context: A212811 A338854 A330957 * A327192 A351651 A157813

Adjacent sequences: A352921 A352922 A352923 * A352925 A352926 A352927

Keyword

sign,base

Author

N. J. A. Sloane, Apr 26 2022

Status

approved