A056154 -id:A056154 - cp's OEIS frontend

A056734 Positive numbers k such that, in base 3, 2^k and 2^(k+1) have the same number of digits and the same number of 0's.

2, 5, 8, 10, 18, 21, 27, 29, 35, 40, 62, 67, 83, 92, 138, 146, 165, 184, 298, 346, 428, 487, 666, 750, 785, 929, 937, 1064, 1086, 1156, 1162, 1240, 1614, 1706, 1739, 1788, 2327, 2389, 2533, 2649, 2937, 3240, 3403, 3489, 3549, 3619, 3693, 3817, 3866, 4175

Offset: 1

Russell Harper (rharper(AT)intouchsurvey.com), Aug 13 2000

Using empirical data for 1 <= k <= 10000, it has been found that the distribution of these terms correlates well (R^2 = 0.9513) with f(k) = c*k^(1/2) with c approximately 0.73. In addition, f'(k) approximates the probability that any particular k has this property. Any terms in A056154 must also be in this sequence.

			First term: 2^2 = 11_3, 2^3 = 22_3, both with 0 zeros and both of length 2.
Second term: 2^5 = 1012_3, 2^6 = 2101_3, both with 1 zero and both of length 4.

Harvey P. Dale, Table of n, a(n) for n = 1..200

Cf. A056154, A117630.

Select[Range[4200],IntegerLength[2^#,3]==IntegerLength[2^(#+1),3] && DigitCount[ 2^#,3,0]==DigitCount[2^(#+1),3,0]&] (* Harvey P. Dale, Dec 10 2021 *)

isok(k) = my(da=digits(2^k, 3), db=digits(2^(k+1), 3)); (#da == #db) && (#select(x->(x==0), da) == #select(x->(x==0), db)); \\ Michel Marcus, Jul 01 2021

A056735 Numbers k such that the base-3 expansions of 2^k and 2^(k+1) have the same number of 1's and the same number of digits.

Original entry on oeis.org

5, 27, 32, 40, 54, 92, 135, 138, 151, 159, 167, 176, 189, 281, 284, 319, 401, 503, 718, 723, 734, 820, 929, 1035, 1086, 1127, 1311, 1341, 1371, 1693, 1785, 1869, 1948, 2010, 2181, 2408, 2563, 2771, 2923, 2983, 3004, 3007, 3210, 3213, 3479, 3527, 4037

Offset: 1

Russell Harper (rharper(AT)intouchsurvey.com), Aug 13 2000

Using empirical data for 1 <= k <= 10000, it has been found that the distribution of these terms correlates well (R^2 = 0.9798) with g(k) = b*sqrt(k) where b ~ 0.70. In addition, g'(k) approximates the probability that any particular k has this property. A056154 is a subsequence.

			a(1)=5: 2^5 = 1012_3, 2^6 = 2101_2, both with two 1's and both of length 4.
a(2)=27: 2^27 = 100100112222002222_3, 2^28 = 200201002221012221_3, both with four 1's and both of length 18.

Cf. A007089, A056154.

Select[Range[4100],Length[IntegerDigits[2^#,3]]==Length[ IntegerDigits[ 2^(#+1),3]] && DigitCount[2^#,3,1]==DigitCount[2^(#+1),3,1]&] (* Harvey P. Dale, Jul 09 2021 *)

A056736 Numbers n such that 2^n in base 3 has same number of 2's as 2^(n+1) in base 3 and 2^n and 2^(n+1) have the same number of digits in base 3.

Original entry on oeis.org

5, 16, 27, 40, 65, 92, 124, 138, 143, 265, 368, 457, 476, 501, 634, 707, 839, 842, 848, 929, 1013, 1086, 1289, 1303, 1587, 1685, 1812, 1926, 1994, 2213, 2308, 2522, 2565, 2950, 3286, 3674, 3774, 3942, 4034, 4318, 4381, 4438, 4719, 4728, 4909, 4971

Offset: 1

Russell Harper (rharper(AT)intouchsurvey.com), Aug 13 2000

Using empirical data for 1 <= n <= 10000, it has been found that the distribution of these terms correlates well (R^2 = 0.9936) to h(n) = c*n^(1/2) with 'c' a constant approximately 0.64. In addition, h'(n) approximates the probability that any particular n has this property. Any terms in sequence A056154 must also satisfy this sequence.

			First term: 2^5 = 1012, 2^6 = 2101, both with 1 two and both of length 4. Second term: 2^16 = 10022220021, 2^17 = 20122210112, both with 5 twos and both of length 11.

A056154.

sn2Q[n_]:=Module[{a=2^n,b=2^(n+1)},DigitCount[a,3,2]==DigitCount[b,3,2] && IntegerLength[a,3]==IntegerLength[b,3]]; Select[Range[5000],sn2Q] (* Harvey P. Dale, Aug 27 2012 *)

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A056734 Positive numbers k such that, in base 3, 2^k and 2^(k+1) have the same number of digits and the same number of 0's.

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A056735 Numbers k such that the base-3 expansions of 2^k and 2^(k+1) have the same number of 1's and the same number of digits.

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A056736 Numbers n such that 2^n in base 3 has same number of 2's as 2^(n+1) in base 3 and 2^n and 2^(n+1) have the same number of digits in base 3.

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