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A309079 For any n > 0: consider the strictly increasing finite sequences of integers whose concatenation of terms, in binary and without leading zeros, equals that of n; a(n) is the minimal sum of the terms of such a finite sequence.

%I A309079 #12 Jul 11 2019 19:14:16
%S A309079 1,2,3,4,5,3,4,8,9,10,5,5,6,7,8,16,17,18,19,6,7,8,9,9,10,11,6,7,8,9,
%T A309079 10,32,33,34,35,36,9,10,11,10,11,12,13,14,15,11,12,17,18,19,20,7,8,9,
%U A309079 10,11,12,13,14,8,9,10,11,64,65,66,67,68,69,70,71,12
%N A309079 For any n > 0: consider the strictly increasing finite sequences of integers whose concatenation of terms, in binary and without leading zeros, equals that of n; a(n) is the minimal sum of the terms of such a finite sequence.
%C A309079 Any integer appear in the sequence:
%C A309079 - for any m > 0 with binary expansion Sum_{k >= 0} b_k * 2^k,
%C A309079 - let n = (Sum_{k >= 0} b_k * 2^Sum_{j >= k} ((1+j) * b_j))/2,
%C A309079 - then a(n) = m,
%C A309079 - for example (in binary): a("1101000") = "1" + "10" + "1000" = "1011".
%H A309079 Rémy Sigrist, <a href="/A309079/b309079.txt">Table of n, a(n) for n = 1..8192</a>
%H A309079 Rémy Sigrist, <a href="/A309079/a309079.gp.txt">PARI program for A309079</a>
%F A309079 a(n) <= n with equality iff n is a power of two or the binary concatenation of 2^k and m for some k >= 0 and m <= 2^k.
%F A309079 a(2*n) <= 2*a(n).
%F A309079 a(2*n + 1) <= 2*a(n) + 1.
%F A309079 a(A164894(k)) = A000225(k) for any k > 0.
%e A309079 The first terms, alongside the corresponding finite sequences, are:
%e A309079   n   a(n)  bin(n)  bin(seq)
%e A309079   --  ----  ------  --------
%e A309079    1     1       1  (1)
%e A309079    2     2      10  (10)
%e A309079    3     3      11  (11)
%e A309079    4     4     100  (100)
%e A309079    5     5     101  (101)
%e A309079    6     3     110  (1,10)
%e A309079    7     4     111  (1,11)
%e A309079    8     8    1000  (1000)
%e A309079    9     9    1001  (1001)
%e A309079   10    10    1010  (1010)
%e A309079   11     5    1011  (10,11)
%e A309079   12     5    1100  (1,100)
%e A309079   13     6    1101  (1,101)
%e A309079   14     7    1110  (1,110)
%e A309079   15     8    1111  (1,111)
%e A309079   16    16   10000  (10000)
%e A309079   17    17   10001  (10001)
%e A309079   18    18   10010  (10010)
%e A309079   19    19   10011  (10011)
%e A309079   20     6   10100  (10,100)
%e A309079   21     7   10101  (10,101)
%o A309079 (PARI) See Links section.
%Y A309079 Cf. A000225, A143789, A164894.
%K A309079 nonn,base
%O A309079 1,2
%A A309079 _Rémy Sigrist_, Jul 11 2019