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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.
2
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
1,1
COMMENTS
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.
LINKS
EXAMPLE
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.
MATHEMATICA
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 *)
PROG
(PARI) 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
CROSSREFS
Sequence in context: A169922 A157481 A100809 * A019995 A188802 A031141
KEYWORD
easy,nonn,base
AUTHOR
Russell Harper (rharper(AT)intouchsurvey.com), Aug 13 2000
STATUS
approved