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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. 1

%I #12 Jul 11 2019 19:14:16

%S 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 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 10,11,12,13,14,8,9,10,11,64,65,66,67,68,69,70,71,12

%N 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 Any integer appear in the sequence:

%C - for any m > 0 with binary expansion Sum_{k >= 0} b_k * 2^k,

%C - let n = (Sum_{k >= 0} b_k * 2^Sum_{j >= k} ((1+j) * b_j))/2,

%C - then a(n) = m,

%C - for example (in binary): a("1101000") = "1" + "10" + "1000" = "1011".

%H Rémy Sigrist, <a href="/A309079/b309079.txt">Table of n, a(n) for n = 1..8192</a>

%H Rémy Sigrist, <a href="/A309079/a309079.gp.txt">PARI program for A309079</a>

%F 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 a(2*n) <= 2*a(n).

%F a(2*n + 1) <= 2*a(n) + 1.

%F a(A164894(k)) = A000225(k) for any k > 0.

%e The first terms, alongside the corresponding finite sequences, are:

%e n a(n) bin(n) bin(seq)

%e -- ---- ------ --------

%e 1 1 1 (1)

%e 2 2 10 (10)

%e 3 3 11 (11)

%e 4 4 100 (100)

%e 5 5 101 (101)

%e 6 3 110 (1,10)

%e 7 4 111 (1,11)

%e 8 8 1000 (1000)

%e 9 9 1001 (1001)

%e 10 10 1010 (1010)

%e 11 5 1011 (10,11)

%e 12 5 1100 (1,100)

%e 13 6 1101 (1,101)

%e 14 7 1110 (1,110)

%e 15 8 1111 (1,111)

%e 16 16 10000 (10000)

%e 17 17 10001 (10001)

%e 18 18 10010 (10010)

%e 19 19 10011 (10011)

%e 20 6 10100 (10,100)

%e 21 7 10101 (10,101)

%Y Cf. A000225, A143789, A164894.

%K nonn,base

%O 1,2

%A _Rémy Sigrist_, Jul 11 2019

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