A306607 -id:A306607 - cp's OEIS frontend

A306740 Numbers k such that A306607(k) = 0.

0, 3, 7, 9, 15, 31, 33, 45, 51, 63, 127, 129, 153, 165, 189, 195, 219, 231, 255, 411, 435, 511, 513, 561, 585, 633, 645, 693, 717, 765, 771, 819, 843, 891, 903, 951, 975, 1023, 2047, 2049, 2145, 2193, 2289, 2313, 2409, 2457, 2553, 2565, 2661, 2709, 2805, 2829, 2925, 2973, 3069, 3075, 3171, 3219

Offset: 1

Robert Israel, Mar 07 2019

The first even terms are 0, 68690167808, 68690561024, 68690757632, 68691150848, 68698560512, 68698953728, 68699150336, 68699543552, 68715331584, 68715724800, 68715921408, 68716314624. - Robert Israel, Mar 10 2019

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A306607. Includes A048701.

f:= proc(n) local L;
  L:= convert(n, base, 2);
  while nops(L) > 1 do
    L:= L[2..-1]-L[1..-2]
  od;
  op(L)
end proc:
select(f=0, [$0..10000]);

seqQ[n_] := NestWhile[Differences, Reverse[IntegerDigits[n, 2]], Length[#]>1&] == {0}; Select[Range[0, 3000], seqQ] (* Amiram Eldar, Mar 08 2019 *)

A306754 The bottom entry in the difference table of the positions of the ones in the binary representation of n.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 1, 0, 3, 3, 2, 1, 1, -1, 0, 0, 4, 4, 3, 2, 2, 0, 1, 1, 1, -2, -1, -2, 0, 1, 0, 0, 5, 5, 4, 3, 3, 1, 2, 2, 2, -1, 0, -1, 1, 2, 1, 1, 1, -3, -2, -4, -1, -1, -2, -3, 0, 2, 1, 3, 0, -1, 0, 0, 6, 6, 5, 4, 4, 2, 3, 3, 3, 0, 1, 0, 2, 3, 2, 2, 2, -2

Offset: 0

Rémy Sigrist, Mar 08 2019

By convention, a(0) = 0.
For any n > 0:
- let (b_1, b_2, ..., b_h) be the positions of the ones in the binary representation of n,
- h = A000120(n) and 0 <= b_1 < b_2 < ... < b_h,
- n = Sum_{k = 1..h} 2^b_k,
- a(n) is the unique value remaining after taking successively the first differences of (b_1, ..., b_h) h-1 times.

			For n = 59:
- the binary representation of 59 is "111011",
- so h = 5 and b_1 = 0, b_2 = 1, b_3 = 3, b_4 = 4, b_5 = 5,
- the corresponding difference table is:
        0   1   3   4   5
          1   2   1   1
            1  -1   0
             -2   1
                3
- hence a(59) = 3.

Rémy Sigrist, Table of n, a(n) for n = 0..16384

See A306607 for a similar sequence.

Cf. A000120, A133457, A209229.

a(n) = { my (h=hammingweight(n), o=0, v=0); forstep (k=h-1, 0, -1, my (w=valuation(n, 2)); o += w; v += (-1)^k * binomial(h-1, k) * o; o++; n\=2^(1+w)); v };

a(2^k) = k for any k >= 0.
a(2^k-1) = [k=2].
a(2*n) = a(n) + A209229(n).

A309776 Form a triangle: first row is n in base 2, next row is sums of pairs of adjacent digits of previous row, repeat until get a single number which is a(n).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 4, 1, 2, 4, 5, 4, 5, 7, 8, 1, 2, 5, 6, 7, 8, 11, 12, 5, 6, 9, 10, 11, 12, 15, 16, 1, 2, 6, 7, 11, 12, 16, 17, 11, 12, 16, 17, 21, 22, 26, 27, 6, 7, 11, 12, 16, 17, 21, 22, 16, 17, 21, 22, 26, 27, 31, 32, 1, 2, 7, 8, 16, 17, 22, 23, 21, 22

Offset: 0

Cameron Musard, Aug 16 2019

a(n) = 1 occurs at n = 2^k for nonnegative integers k.

			For n=5 the triangle is
  1 0 1
   1 1
    2
so a(5)=2.
For n=14 we get
  1 1 1 0
   2 2 1
    4 3
     7
so a(14)=7.
For n=26=11010_2; (n1+n2, n2+n3, n3+n4, n4+n5) = 2111; (n1'+n2', n2'+n3', n3'+n4') = 322; (n1''+n2'', n2''+n3'') = 54; (n1'''+n2''') = 9; a(26)= 9.

Rémy Sigrist, Table of n, a(n) for n = 0..16384

Cf. A306607.

a(n) = my (b=binary(n)); sum(k=1, #b, b[k]*binomial(#b-1,k-1)) \\ Rémy Sigrist, Aug 20 2019

From Bernard Schott, Sep 22 2019: (Start)
a(2^k + 1) = 2 for k >= 1 where 2^k+1 = 1000..0001_2.
a(2^k - 1) = 2^(k-1) for k >= 2 where 2^k-1 = 111..111_2.
a((4^k-1)/3) = 2^(2*k-3) for k >= 2 where (4^k-1)/3 = 10101..0101_2.
(End)

Edited by N. J. A. Sloane, Sep 21 2019

Data corrected by Rémy Sigrist, Sep 22 2019

cp's OEIS Frontend

A306740 Numbers k such that A306607(k) = 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A306754 The bottom entry in the difference table of the positions of the ones in the binary representation of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A309776 Form a triangle: first row is n in base 2, next row is sums of pairs of adjacent digits of previous row, repeat until get a single number which is a(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions