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A322827 A permutation of A025487: Sequence of least representatives of distinct prime signatures obtained from the run lengths present in the binary expansion of n.

%I A322827 #41 Feb 05 2020 16:54:21
%S A322827 1,2,6,4,36,30,12,8,216,180,210,900,72,60,24,16,1296,1080,1260,5400,
%T A322827 44100,2310,6300,27000,432,360,420,1800,144,120,48,32,7776,6480,7560,
%U A322827 32400,264600,13860,37800,162000,9261000,485100,30030,5336100,1323000,69300,189000,810000,2592,2160,2520,10800,88200,4620,12600
%N A322827 A permutation of A025487: Sequence of least representatives of distinct prime signatures obtained from the run lengths present in the binary expansion of n.
%C A322827 A101296(a(n)) gives a permutation of natural numbers.
%H A322827 Antti Karttunen, <a href="/A322827/b322827.txt">Table of n, a(n) for n = 0..16383</a>
%H A322827 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A322827 a(n) = A046523(a(n)) = A046523(A322825(n)).
%F A322827 A001221(a(n)) = A005811(n).
%F A322827 A001222(a(n)) = A227183(n).
%F A322827 A322585(a(n)) = 1.
%e A322827 The sequence can be represented as a binary tree:
%e A322827                                       1
%e A322827                                       |
%e A322827                    ...................2...................
%e A322827                   6                                       4
%e A322827        36......../ \........30                 12......../ \........8
%e A322827        / \                 / \                 / \                 / \
%e A322827       /   \               /   \               /   \               /   \
%e A322827      /     \             /     \             /     \             /     \
%e A322827    216      180         210    900         72       60         24       16
%e A322827 etc.
%e A322827 Both children are multiples of their common parent, see A323503, A323504 and A323507.
%e A322827 The value of a(n) is computed from the binary expansion of n as follows: Starting from the least significant end of the binary expansion of n (A007088), we record the successive run lengths, subtract one from all lengths except the first one, and use the reversed partial sums of these adjusted values as the exponents of successive primes.
%e A322827 For 11, which is "1011" in base 2, we have run lengths [2, 1, 1] when scanned from the right, and when one is subtracted from all except the first, we have [2, 0, 0], partial sums of which is [2, 2, 2], which stays same when reversed, thus a(11) = 2^2 * 3^2 * 5^2 = 900.
%e A322827 For 13, which is "1101" in base 2, we have run lengths [1, 1, 2] when scanned from the right, and when one is subtracted from all except the first, we have [1, 0, 1], partial sums of which is [1, 1, 2], reversed [2, 1, 1], thus a(13) = 2^2 * 3^1 * 5^1 = 60.
%e A322827 Sequence A227183 is based on the same algorithm.
%t A322827 {1}~Join~Array[Times @@ MapIndexed[Prime[First@ #2]^#1 &, Reverse@ Accumulate@ MapIndexed[Length[#1] - Boole[First@ #2 > 1] &, Split@ Reverse@ IntegerDigits[#, 2]]] &, 54] (* _Michael De Vlieger_, Feb 05 2020 *)
%o A322827 (PARI) A322827(n) = if(!n,1,my(bits = Vecrev(binary(n)), rl=1, o = List([])); for(i=2,#bits,if(bits[i]==bits[i-1], rl++, listput(o,rl))); listput(o,rl); my(es=Vecrev(Vec(o)), m=1); for(i=1,#es,m *= prime(i)^es[i]); (m));
%Y A322827 Cf. A000079 (right edge), A000400 (left edge, apart from 2), A005811, A046523, A101296, A227183, A322585, A322825, A323503, A323504, A323507.
%Y A322827 Other rearrangements of A025487 include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181821, A181822.
%Y A322827 Cf. A005940, A283477, A323505 for other similar trees.
%K A322827 nonn,tabf,look
%O A322827 0,2
%A A322827 _Antti Karttunen_, Jan 16 2019