A104321 Smallest number m such that A104320(m)=n.
0, 5, 8, 18, 13, 26, 27, 23, 42, 25, 37, 58, 47, 46, 61, 67, 54, 71, 77, 73, 88, 99, 141, 100, 115, 114, 119, 117, 113, 112, 109, 135, 110, 127, 133, 136, 164, 162, 177, 186, 193, 195, 163, 189, 201, 196, 191, 199, 206, 188, 208, 200, 221, 266, 235, 234, 238, 280
Offset: 0
Keywords
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
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, ...
Examples
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.
References
- Ilan Vardi, "Computational Recreations in Mathematica," Addison-Wesley Publishing Co., Redwood City, CA, 1991, page 20.
Links
- Brian Hayes, Third Base, November-December 2001, Volume 89, Number 6, Page 490.
Programs
-
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
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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 *) -
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
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
A365214 Least k such that the binary representation of 3^k has exactly n 1's, or -1 if no such k exists.
0, 1, 4, 3, 7, 5, -1, 9, 10, 12, 13, -1, 11, 14, 15, 24, 19, 25, 22, 21, -1, 23
Offset: 1
Comments
Least k such that A011754(k) = n, or -1 if no such k exists.
The first 22 terms are from Dimitrov and Howe (2021). After a(22), the sequence continues (with the -1's conjectural but very likely correct) -1, 26, 28, 31, 36, 40, 34, 27, -1, 35, 49, 33, 53, 38, -1, 42, 41, 57, 48, 52, -1, -1, 46, 51, 67, 59, 62, -1, ... .
Links
- Vassil S. Dimitrov and Everett W. Howe, Powers of 3 with few nonzero bits and a conjecture of Erdős, arXiv:2105.06440 [math.NT], 2021.
Comments
Links
Crossrefs
Programs
PARI
a(n) = {my(k = 0); while (#select(x->(x==0), digits(2^k, 3)) != n, k++); k;} \\ Michel Marcus, Oct 19 2016