A011754 -id:A011754 - cp's OEIS frontend

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

Pontus von Brömssen, Aug 26 2023

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, ... .

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.

Cf. A011754, A364650, A365215, A375472.

A369857 Number of nonzero bits (a.k.a. binary or Hamming weight) of 7^n.

Original entry on oeis.org

1, 3, 3, 6, 5, 7, 9, 11, 14, 17, 15, 20, 11, 20, 23, 20, 22, 24, 20, 31, 30, 24, 24, 36, 29, 29, 37, 33, 37, 42, 43, 43, 40, 50, 53, 44, 39, 53, 43, 60, 57, 53, 60, 59, 62, 68, 65, 66, 59, 68, 75, 71, 84, 77, 65, 74, 81, 87, 85, 83, 77, 89, 83, 95, 84, 89, 86, 98, 102, 94, 97, 104

Offset: 0

M. F. Hasler, Apr 17 2024

nonn

Conjecture: a(n)/n -> log_4(7) = 1.403677461..., i.e., about half of the bits of 7^n are nonzero.

			The first few powers of 7 and their binary representation are as follows:
   n  | 0 |  1  |    3   |     4     |       5      |        6        | ...
  ----+---+-----+--------+-----------+--------------+-----------------+----
  7^n | 1 |  7  |   49   |    343    |     2401     |      16807      | ...
  ----+---+-----+--------+-----------+--------------+-----------------+----
  bin | 1 | 111 | 110001 | 101010111 | 100101100001 | 100000110100111 | ...
  ----+---+-----+--------+-----------+--------------+-----------------+----
  a(n)| 1 |  3  |    3   |     6     |       5      |        7        | ...
  ----+---+-----+--------+-----------+--------------+-----------------+----

Hugo Pfoertner, Plot of a(n) - 1.40368*n, +-4*sqrt(n), n up to 10^6.

Cf. A000120 (Hamming weight), A000420 (7^n).

Cf. A011754, A118738 (analog for 3^n and 5^n).

Maple
```
A369857 := A000120 @ A000420
```

apply( {A369857(n) = hammingweight(7^n)}, [0..99])

A369857 = lambda n: (7**n).bit_count() # for Python >= 3.10

a(n) = A000120(A000420(n)). (Definition of this sequence.)

A371970 Exponents k such that the binary expansion of 3^k has an even number of ones.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 12, 14, 17, 18, 21, 23, 24, 25, 26, 27, 31, 32, 33, 35, 37, 38, 39, 40, 42, 44, 45, 47, 51, 52, 55, 57, 58, 59, 60, 61, 64, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 96, 99, 102, 104, 105, 106, 109, 112, 116, 127, 131, 132, 133, 134, 135, 136

Offset: 1

Hugo Pfoertner, Apr 24 2024

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Complement of A223024.

Cf. A000120, A000244, A001969, A011754, A078839.

q:= n-> is(add(i, i=Bits[Split](3^n))::even):
select(q, [$0..150])[];  # Alois P. Heinz, Apr 24 2024

Select[Range[136], EvenQ@ DigitCount[3^#, 2, 1] &] (* Michael De Vlieger, Apr 24 2024 *)

is_a371970(k) = hammingweight(3^k)%2 == 0

A093476 Index of occurrence of the first 0 bit in binary representation of 3^n.

Original entry on oeis.org

2, 3, 2, 5, 2, 2, 3, 2, 4, 2, 2, 3, 2, 3, 2, 5, 2, 2, 3, 2, 4, 2, 2, 3, 2, 3, 2, 6, 2, 2, 3, 2, 4, 2, 2, 3, 2, 4, 2, 7, 2, 2, 3, 2, 5, 2, 2, 3, 2, 4, 2, 2, 3, 2, 3, 2, 5, 2, 2, 3, 2, 4, 2, 2, 3, 2, 3, 2, 5, 2, 2, 3, 2, 4, 2, 2, 3, 2, 3, 2, 6, 2, 2, 3, 2, 4, 2, 2, 3, 2, 4, 2, 7, 2, 2, 3, 2, 5, 2, 2, 3, 2, 4, 2, 2

Offset: 2

Benoit Cloitre, May 22 2004

nonn

			In binary, 3^5 = [1, 1, 1, 1, 0, 0, 1, 1] where the first 0 occurs at 5th place. Hence a(5)=5.

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A011754, A048651.

seq(ListTools:-Search(0, ListTools:-Reverse(convert(3^n,base,2))), n=2..200); # Robert Israel, Nov 20 2017

Array[FirstPosition[IntegerDigits[3^#, 2], 0][[1]] &, 105, 2] (* Michael De Vlieger, Nov 20 2017 *)

a(n)=if(n<2,0,s=1;while(component(binary(3^n),s)>0,s++);s)

It seems that Sum_{i=2..n} a(i) is asymptotic to c*n with c=2.7(8).....
From Robert Israel, Nov 20 2017: (Start)
a(n) = k if log_2(2 - 1/2^(k-2)) < frac(n*log_2(3)) < log_2(2 - 1/2^(k-1)). By the equidistribution theorem, this occurs with asymptotic density log_2(2-1/2^(k-1)) - log_2(2-1/2^(k-2)).
Thus c = Sum_{k>=2} k (log_2(2-1/2^(k-1)) - log_2(2 - 1/2^(k-2))) = 2 - Sum_{k>=2} log_2(1-1/2^k) = 2.791916824662...  Note that A048651 is the decimal expansion of 2^(1-c). (End)

A365215 Largest k such that the binary representation of 3^k has exactly n 1's, or -1 if no such k exists.

Original entry on oeis.org

0, 2, 4, 3, 7, 8, -1, 9, 10, 12, 16, -1, 11, 18, 15, 24, 20, 25, 22, 21, -1, 23

Offset: 1

Pontus von Brömssen, Aug 26 2023

Largest k such that A011754(k) = n, or -1 if no such k exists.
Senge and Straus prove that a(n) is finite for all n.
The first 22 terms are from Dimitrov and Howe (2021). After a(22), the sequence conjecturally but very likely continues -1, 26, 30, 32, 36, 40, 34, 27, -1, 39, 49, 45, 53, 38, -1, 47, 56, 57, 50, 58, -1, -1, 66, 51, 67, 59, 62, -1, ... .

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.
H. G. Senge and E. G. Straus, PV-numbers and sets of multiplicity, Periodica Mathematica Hungarica 3 (1973), 93-100.

Cf. A011754, A364650, A365214.

LargestK[n_Integer] := Module[{k = 1000(*Assuming 1000 is large enough for the search.  Adjust if necessary.*), binCount}, While[k >= 0, binCount = Total[IntegerDigits[3^k, 2]]; If[binCount == n, Return[k]]; k--;]; -1]; Table[LargestK[n], {n, 22}] (* Robert P. P. McKone, Aug 26 2023 *)

A371971 a(n) is the least exponent m minimizing the percentage of set bits in the binary representation of prime(n)^m.

Original entry on oeis.org

24, 25, 12, 6, 16, 2, 10, 2, 7, 22, 16, 8, 15, 2, 14, 3, 20, 2, 4, 7, 21, 4, 19, 6, 13, 6, 7, 7, 3, 10, 2, 1, 9, 5, 2, 2, 6, 10, 36, 11, 13, 2, 2, 10, 9, 5, 2, 34, 5, 2, 4, 4, 8, 2, 9, 4, 6, 9, 3, 14, 38, 9, 14, 16, 2, 7, 4, 16, 3, 9, 6, 2, 4, 2, 10, 15, 10, 14, 6, 71, 31, 4

Offset: 2

Hugo Pfoertner, Apr 17 2024

Hugo Pfoertner, Logfile of downward searches for {3, 5, 7, 11, 13}^m, m = 10^6 .. 1.
Hugo Pfoertner, Plot of Hammingweight(3^m)/log_2(3^m), m=2..645.

Cf. A000120, A070939, A011754, A118738, A369857.

a371971(n,limit=10000) = {my(p=prime(n), q=1, m=1); for (k=1,limit, my (pp=p^k, h=hammingweight(pp), g=#binary(pp), qq=h/g); if (qq

A375258 Array read by antidiagonals: T(k,n) is the least positive integer whose sum of base-2 digits is k and sum of base-3 digits is n, or -1 if there is none.

Original entry on oeis.org

1, 2, 3, -1, 6, 81, 8, 5, 28, 27, -1, 20, 7, 30, 2187, 128, 17, 14, 15, 244, 243, -1, 68, 25, 46, 31, 246, -1, 512, 8193, 26, 23, 94, 63, 6570, 19683, -1, 80, 131, 78, 47, 126, 247, 2430, 59049, 2048, 1025, 134, 53, 62, 95, 254, 255, 19926, 531441, -1, 2050, 161, 212, 79, 222, 127, 766, 2431

Offset: 1

Robert Israel, Aug 07 2024

T(k,n) is the least positive integer x, if it exists, such that A000120(x) = k and A053735(x) = n.
T(k,n) == n (mod 2) unless T(k,n) = -1, since A053735(x) == x (mod 2). In particular, T(1, n) = -1 if n >= 3 is odd.
Dimitrov and Howe prove that for n > 25, the sum of binary digits of 3^n is > 22.  In particular, this implies T(7,1) = T(12,1) = T(21,1) = -1, since none of the first 25 powers of 3 work.

			Array starts
     1,     2,    -1,     8,    -1,   128,    -1,   512, ...
     3,     6,     5,    20,    17,    68,  8193,    80, ...
    81,    28,     7,    14,    25,    26,   131,   134, ...
    27,    30,    15,    46,    23,    78,    53,   212, ...
  2187,   244,    31,    94,    47,    62,    79,   158, ...
   243,   246,    63,   126,    95,   222,   125,   238, ...
    -1,  6570,   247,   254,   127,   382,   223,   446, ...
 19683,  2430,   255,   766,   507,   510,   383,   958, ...

Robert Israel, Table of n, a(n) for n = 1..253 (first 22 antidiagonals)
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.

Cf. A000120, A011754, A037301, A053735, A375257.

T:= Matrix(8,8,-1):
for x from 1 to 10^5 do
  k:= convert(convert(x,base,2),`+`);
  n:= convert(convert(x,base,3),`+`);
  if k <= 8 and n <= 8 and T[k,n] = -1 then T[k,n]:= x; fi
od:
T;

A070972 Length of longest run of consecutive 1's in binary expansion of 3^n (A004656).

Original entry on oeis.org

1, 2, 1, 2, 1, 4, 2, 2, 2, 3, 3, 7, 6, 4, 5, 4, 3, 4, 4, 4, 6, 5, 5, 4, 3, 2, 10, 8, 7, 5, 4, 5, 7, 9, 8, 6, 5, 7, 11, 10, 8, 6, 6, 4, 9, 7, 6, 4, 9, 8, 6, 9, 8, 6, 5, 5, 3, 4, 7, 5, 10, 8, 7, 6, 6, 6, 6, 4, 4, 7, 7, 5, 5, 5, 5, 5, 5, 9, 8, 6, 5, 6, 5, 6, 5, 5, 4, 6, 5, 10, 8, 7, 5, 6, 6, 6, 7, 7, 6, 7, 5, 5, 5, 9

Offset: 0

Frank Schwellinger (nummer_eins(AT)web.de), Jan 03 2003

			3^11 = (101011001111111011) binary, so a(11) = 7.

Cf. A004656 and A011754.

f[n_Integer] := Block[{p = Flatten[ Position[ Prepend[ IntegerDigits[2*3^n, 2], 0], 0]]}, Max[Drop[p, 1] - Drop[p, -1]] - 1]; Table[ f[n], {n, 0, 103}]

Edited by Robert G. Wilson v, Jan 04 2002

A118736 Number of zeros in binary expansion of 3^n.

Original entry on oeis.org

0, 0, 2, 1, 4, 2, 4, 7, 7, 7, 7, 5, 10, 10, 9, 9, 15, 13, 15, 14, 15, 14, 16, 15, 23, 22, 18, 13, 20, 21, 23, 24, 25, 19, 25, 24, 31, 25, 25, 30, 36, 26, 29, 30, 36, 38, 28, 37, 36, 45, 39, 35, 41, 50, 47, 46, 50, 51, 50, 46, 40, 41, 50, 43, 46, 53, 60, 60, 53, 55, 47, 45, 57, 58

Offset: 0

Zak Seidov, May 22 2006

Taylor Dupuy, David E. Weirich, Bits of 3^n in binary, Wieferich primes and a conjecture of Erdős, Journal of Number Theory, Volume 158, January 2016, Pages 268-280.

Cf. A011754 (number of ones in binary expansion of 3^n).

f[n_] := DigitCount[3^n, 2, 0]; Table[f[n], {n, 0, 73}] (* Ray Chandler, Sep 29 2006 *)

a(n) = #select(x->(x==0), binary(3^n)); \\ Michel Marcus, Apr 14 2020

cp's OEIS Frontend

A365214 Least k such that the binary representation of 3^k has exactly n 1's, or -1 if no such k exists.

Views

Author

Keywords

Comments

Links

Crossrefs

A369857 Number of nonzero bits (a.k.a. binary or Hamming weight) of 7^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Python

Formula

A371970 Exponents k such that the binary expansion of 3^k has an even number of ones.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A093476 Index of occurrence of the first 0 bit in binary representation of 3^n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A365215 Largest k such that the binary representation of 3^k has exactly n 1's, or -1 if no such k exists.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A371971 a(n) is the least exponent m minimizing the percentage of set bits in the binary representation of prime(n)^m.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A375258 Array read by antidiagonals: T(k,n) is the least positive integer whose sum of base-2 digits is k and sum of base-3 digits is n, or -1 if there is none.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A070972 Length of longest run of consecutive 1's in binary expansion of 3^n (A004656).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Extensions

A118736 Number of zeros in binary expansion of 3^n.