A275536 Differences of the exponents of the adjacent distinct powers of 2 in the binary representation of n (with -1 subtracted from the least exponent present) are concatenated as decimal digits in reverse order.

1, 2, 11, 3, 12, 21, 111, 4, 13, 22, 112, 31, 121, 211, 1111, 5, 14, 23, 113, 32, 122, 212, 1112, 41, 131, 221, 1121, 311, 1211, 2111, 11111, 6, 15, 24, 114, 33, 123, 213, 1113, 42, 132, 222, 1122, 312, 1212, 2112, 11112

Offset: 1

Author

Armands Strazds, Aug 01 2016

Keywords

• nonn
• base
• fini

Comments

A preferable representation is a sequence of arrays, since multi-digit items are possible: [1],[2],[1,1],[3],[1,2],[2,1],[1,1,1],[4],[1,3],[2,2],[1,1,2],[3,1],[1,2,1],[2,1,1],[1,1,1,1],[5],[1,4],[2,3],[1,1,3],[3,2],[1,2,2],[2,1,2],[1,1,1,2],[4,1],[1,3,1],[2,2,1],[1,1,2,1],[3,1,1],[1,2,1,1],[2,1,1,1],[1,1,1,1,1],[6],[1,5],[2,4],[1,1,4],[3,3],[1,2,3],[2,1,3],[1,1,1,3],[4,2],[1,3,2],[2,2,2],[1,1,2,2],[3,1,2],[1,2,1,2],[2,1,1,2],[1,1,1,1,2]. 0 is not allowed as a digit.

a(512) is the first term which cannot be expressed unambiguously in decimal. - Charles R Greathouse IV, Aug 02 2016

The first two terms which are equal (because of the ambiguity inherent in using decimal, or more generally any finite base) are a(3) = a(1024) = 11. a(3) corresponds to the array [1,1] while a(1024) corresponds to [11]. - Charles R Greathouse IV, Mar 19 2017

Examples

			5 = 2^2 + 2^0, so the representation is [2-0, 0-(-1)] = [2, 1] so a(5) = 12.
6 = 2^2 + 2^1, so the representation is [2-1, 1-(-1)] = [1, 2] so a(6) = 21.
18 = 2^4 + 2^1, so the representation is [4-1, 1-(-1)] = [3, 2] so a(18) = 23.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..511

Crossrefs

Cf. A069800, A261300, A248646, A256494, A258055, A004086, A004719, A071160.

Programs

PARI

a(n)=my(v=List(),k); while(n, k=valuation(n,2)+1; n>>=k; listput(v,k)); fromdigits(Vec(v)) \\ Charles R Greathouse IV, Aug 02 2016

PHP

function dec2delta($k) {
  $p = -1;
  while ($k > 0) {
    $k -= $c = pow(2, floor(log($k, 2)));
    if ($p > -1) $d[] = $p - floor(log($c, 2));
    $p = floor(log($c, 2));
  }
  $d[] = $p + 1;
  return array_reverse($d);
}
function delta2dec($d) {
  $k = 0;
  $e = -1;
  foreach ($d AS $v) {
    if ($v > 0) {
      $e += $v;
      $k += pow(2, $e);
    }
  }
  return $k;
}

Formula

For n=1..511, a(n) = A004086(A004719(A071160(n))) [In other words, terms of A071160 with 0-digits deleted and the remaining digits reversed.] - Antti Karttunen, Sep 03 2016