|
%I #9 Oct 12 2023 14:10:14
%S 1,2,4,3,8,16,9,32,64,5,6,12,27,128,256,81,512,1024,25,18,36,243,2048,
%T 4096,729,8192,16384,7,10,20,15,24,48,45,96,192,125,54,108,2187,32768,
%U 65536,6561,131072,262144,625,162,324,19683,524288,1048576,59049
%N a(n) has as many prime factors as the ternary expansion of n has runs of nonzero digits; if the k-th run corresponds to A032924(e) and appears after m-1 0's then the p-adic valuation of a(n) is e (where p corresponds to the m-th prime number).
%C This sequence is a variant of the Doudna sequence (A005940); here we consider runs of nonzero digits in ternary expansions, there in binary expansions.
%C This sequence is a bijection from the nonnegative integers to the positive integers with inverse A366348.
%C We can devise a similar sequence for any fixed base b >= 2:
%C - the case b = 2 corresponds (up to the offset) to the Doudna sequence (A005940),
%C - the case b = 3 corresponds to the present sequence,
%C - the case b = 10 corresponds to A290389.
%H Rémy Sigrist, <a href="/A366347/a366347.gp.txt">PARI program</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F a(3*n) = A003961(a(n)).
%F a(3^k) = prime(1 + k) for any k >= 0.
%F a(2 * 3^k) = prime(1 + k)^2 for any k >= 0.
%F a(n) is squarefree iff n belongs to A060140.
%e For n = 46: the ternary expansion of 46 is "1201; we have two runs of nonzero digits: "12" (= 5 = A032924(4)) after 2-1 0's and "1" (= 1 = A032924(1)) after 1-1 0's; so a(46) = prime(2)^4 * prime(1)^1 = 3^4 * 2^1 = 162.
%o (PARI) See Links section.
%Y Cf. A003961, A005940, A032924, A060140, A290389, A365278, A366348 (inverse).
%K nonn,base
%O 0,2
%A _Rémy Sigrist_, Oct 07 2023
|
