cp's OEIS Frontend

A233569 Canonical parts power representation of n: n = concatenation((1)^k_1,(10)^k_2,...).

%I A233569 #39 Dec 26 2013 03:20:44
%S A233569 0,1,2,3,4,6,6,7,8,12,10,14,12,14,14,15,16,24,20,28,20,26,26,30,24,28,
%T A233569 26,30,28,30,30,31,32,48,40,56,36,52,52,60,40,52,42,58,52,58,58,62,48,
%U A233569 56,52,60,52,58,58,62,56,60,58,62,60,62,62,63,64,96,80
%N A233569 Canonical parts power representation of n: n = concatenation((1)^k_1,(10)^k_2,...).
%C A233569 Two numbers n_1 and n_2 are called c-equivalent (n_1~n_2) if in the binary they have the same parts of the form 10...0 with k>=0 zeros up to a permutation of them. For example, 6~5, 14~13~11, 12~9.
%C A233569 Denote by (10...0)^k the concatenation k the same consecutive parts (10...0). By agreement, (10...0)^0 denotes the absence of the corresponding part in the binary of n. Let n contains k_i parts with i-1 zeros, i=1,2,... . Then n~concatenation((1)^k_1, (10)^k_2,(100)^k_3,...). The latter number is a(n). Thus a(n_1)=a(n_2) if and only if n_1~n_2. For example, since a(19)=28 which is in binary 11100, then the canonical representation of 19 is (1)^2[*](100), where [*] means concatenation. Analogously, since a(23)=30 which in binary 11110, then the canonical representation of 23 is (1)^3[*](10).
%C A233569 As a natural application, consider a notion of parts power divisor of canonical representation of n. We consider parts power divisors only of the form a(m).
%C A233569 If the canonical representation of n is a(n)=(1)^k_1[*](10)^k_2[*](100)^k_3[*]..., then number a(m) is a parts power divisor  of a(n), iff a(m)=(1)^t_1[*](10)^t_2[*](100)^t_3[*]... with all t_i<=k_i. In particular, 0 (with all t_i=0) is parts power divisor of every a(n). From this it follows that the number of primes power divisors of a(n) is (k_1+1)*(k_2+1)*... This number is an upper estimate for A124771(n).
%H A233569 Peter J. C. Moses, <a href="/A233569/b233569.txt">Table of n, a(n) for n = 0..2499</a>
%H A233569 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%t A233569 bitPatt[n_]:=bitPatt[n]=Split[IntegerDigits[n,2],#1>#2||#2==0&];Map[FromDigits[Flatten[Sort[bitPatt[#]]],2]&,Range[0,33]] (* _Peter J. C. Moses_, Dec 14 2013 *)
%Y A233569 Cf. A114994.
%K A233569 nonn,base
%O A233569 0,3
%A A233569 _Vladimir Shevelev_, Dec 13 2013
%E A233569 More terms from _Peter J. C. Moses_, Dec 15 2013