A060035 Least m >= 0 such that 2^m has n 2's in its base-3 expansion.

0, 1, 3, 12, 9, 16, 15, 19, 27, 30, 44, 40, 55, 52, 65, 60, 51, 75, 73, 80, 86, 82, 81, 77, 98, 85, 95, 79, 118, 141, 162, 107, 129, 105, 158, 145, 155, 143, 138, 152, 203, 176

Offset

0,3

Comments

Previous name was: First power of 2 which has n 2's in its base 3 expansion, or -1 if no such power exists.

"Paul Erdős conjectured that for n > 8, 2^n is not a sum of distinct powers of 3. In terms of digits, this states that powers of 2 for n > 8 must always contain a '2' in their base 3 expansion."

The value of a(42) is conjectured to be -1 because no power of 2 up to 2^10^7 has exactly 42 2's.

After a(42), that is unknown, the sequence goes on 171, 142, 167, 197, 168, 216, 229, 193, 232, 236, 248, 226, 230, 224, 228, 303, 244, ...

References

Ilan Vardi, "Computational Recreations in Mathematica," Addison-Wesley Publishing Co., Redwood City, CA, 1991, page 20.

Links

Table of n, a(n) for n=0..41.

Brian Hayes, Third Base

Example

a(0) = 0 because 2^0 in base 3 is {1} which has no terms equaling 2.
a(6) = 15 because 2^15 in base 3 is {1, 1, 2, 2, 2, 2, 1, 1, 2, 2} which has 6 terms equaling 2.

Maple

for m from 0 to 1000 do
  r:= numboccur(2, convert(2^m, base, 3));
  if not assigned(A[r]) then A[r]:= m fi;
od:
seq(A[i], i=0..41); # Robert Israel, Dec 08 2015

Mathematica

a[n_] := For[k=0, True, k++, If[Count[IntegerDigits[2^k, 3], 2]==n, Return[k]]]; Table[a[n], {n, 0, 41}] (* goes into infinite loop for n > 41 *)
a[n_] := -1; Do[m = Count[IntegerDigits[2^(n), 3], 2]; If[a[m] == -1, a[m] = n], {n, 0, 1000}]; Table[a[n], {n, 0, 59}] {* L. Edson Jeffery, Dec 08 2015 *)

Prog

(PARI) isok(n, k) = {d = digits(2^k, 3); sum(i=1, #d, d[i]==2) == n; }
a(n) = {k = 0; while(! isok(n, k), k++); k; } \\ Michel Marcus, Dec 08 2015

Crossrefs

Sequence in context: A070706 A239932 A114237 * A165988 A298028 A215842

Adjacent sequences: A060032 A060033 A060034 * A060036 A060037 A060038

Keyword

nonn,base,more

Author

Robert G. Wilson v, Mar 17 2001

Extensions

Corrected and extended by Sascha Kurz, Jan 31 2003

Zero prepended to sequence by L. Edson Jeffery, Dec 08 2015

New name from L. Edson Jeffery, Dec 08 2015

a(42) = -1 and following terms removed from data by Michel Marcus, Dec 09 2015

Status

approved