A354757 -id:A354757 - cp's OEIS frontend

A354783 If the binary expansion of A354757(n) is 1 d_1 d_2 ... d_k, then the binary expansion of a(n) is c_1 c_2 ... c_k, where c_i = 1 - d_i.

0, 0, 1, 1, 3, 0, 4, 4, 12, 0, 3, 3, 19, 2, 34, 0, 64, 64, 76, 8, 136, 0, 256, 256, 768, 0, 17, 17, 1041, 16, 50, 32, 2080, 0, 4096, 4096, 12288, 0, 68, 68, 16452, 64, 200, 128, 32896, 0, 65536, 65536, 196608, 0, 768, 768, 262912, 512, 524800, 0, 1048576, 1048576, 1049601, 1024, 2098176, 0, 18, 18, 4194322, 16, 2096, 2048, 8390656, 0, 16777216

Offset: 1

N. J. A. Sloane, Jul 08 2022

Has the same relation to A354757 as A354781 does to A354780.

The offset is 1, to avoid having to define a(0).

			A354757(5) = 12 = 1100_2, so a(5) = 11_2 = 3.
A354757(6) = 15 = 1111_2, so a(6) = 0.
A354757(7) = 27 = 11011_2, so a(7) = 100_2 = 4.

N. J. A. Sloane, Table of n, a(n) for n = 1..4800

Cf. A354169, A354757, A354780, A354781.

See A354793 for Hamming weight of a(n).

Added comment and examples. - N. J. A. Sloane, Aug 02 2022

A354758 a(n) = Product_{k = ceiling(n/2)..n-1} A090252(k).

Original entry on oeis.org

1, 1, 2, 6, 15, 60, 140, 1260, 2772, 36036, 153153, 1225224, 3325608, 76488984, 212469400, 4461857400, 11763078600, 364655436600, 1037865473400, 42552484409400, 107632754682600, 5058739470082200, 33514148989294575, 536226383828713200, 1665124033994425200

Offset: 1

Rémy Sigrist, Jun 06 2022

nonn

The prime divisors of a(n) are forbidden in A090252(n).

			a(7) = A090252(4) * A090252(5) * A090252(6) = 5 * 4 * 7 = 140.

Michael De Vlieger, Table of n, a(n) for n = 1..587
Rémy Sigrist, Representation of the prime divisors of a(n) (blue pixels) and A090252(n) (red pixels)
Rémy Sigrist, PARI program

A355057 is another version.

Cf. A090252, A354757.

With[{s = Import["https://oeis.org/A090252/b090252.txt", "Data"][[1 ;; 25, -1]]}, Table[Product[s[[k]], {k, Ceiling[n/2], n - 1}], {n, Length[s]}]] (* Michael De Vlieger, Aug 28 2022 *)

PARI
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See Links section.
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gcd(a(n), A090252(n)) = 1.

A354780 a(n) is the bitwise OR of (the binary expansions of) b(n+1) to b(2*n), where b is A354169.

Original entry on oeis.org

2, 12, 27, 115, 252, 1004, 2013, 4031, 16307, 32631, 65279, 261375, 524270, 2096110, 4194253, 8386527, 16773119, 67096575, 134217659, 536854459, 1073741623, 2147450751, 4294901759, 17179672575, 34359737599, 137438690559, 274877382143, 549754765311, 2199022205950, 4398044412927, 8796093022189, 35184367894509, 70368744175567

Offset: 1

N. J. A. Sloane, Jul 05 2022

If the binary expansion of a(n) has a 1 in the 2^i's bit (for any i >= 0) then A354169(2*n+1) must have a 0 in that bit.
A354169(2*n+1) is the smallest number not yet in A354169 which satisfies that condition (this follows at once from the definition of A354169).
This sequence bears the same relation to A354169 as A355057 does to A090252.

			Consider n=6. Then b(7) to b(12) are 32, 64, 12, 128, 256, 512. The bitwise OR of those 6 numbers is 1111101100_2 = 1004_10 = a(6). The bitwise complement of 1004_10 is 10011_2 = 19_10 = A354781(6), and A354169(6) = 17_10 = 10001_2.
On the other hand, for n=5, b(6) to b(10) are 16, 32, 64, 12, 128, whose bitwise OR is 11111100_2 = 252_10 = a(5). The bitwise complement of 252_10 is 3_10 = 11_2 = A354781(5). However, 3 has already appeared in A354169, and the smallest available number whose binary expansion is disjoint from 252_10 = 11111100_2 is 2^8 = 100000000_2 = 256_10 = 2^8 = A354169(5).

N. J. A. Sloane, Table of n, a(n) for n = 1..256

A bisection of A354757.
Cf. A354169, A354680, A354757, A354758, A354767, A354773, A354774, A354778, A355150, A354781.
See also A090252, A355057.

A354781 If the binary expansion of A354780(n) is 1 d_1 d_2 ... d_k, then the binary expansion of a(n) is c_1 c_2 ... c_k, where c_i = 1 - d_i.

Original entry on oeis.org

1, 3, 4, 12, 3, 19, 34, 64, 76, 136, 256, 768, 17, 1041, 50, 2080, 4096, 12288, 68, 16452, 200, 32896, 65536, 196608, 768, 262912, 524800, 1048576, 1049601, 2098176, 18, 4194322, 2096, 8390656, 16777216, 50331648, 12288, 67121152, 134225920, 268435456, 268451844, 536887296, 72, 1073741896, 32960, 2147516416, 4294967296, 12884901888

Offset: 1

N. J. A. Sloane, Jul 05 2022

			See A354780.

N. J. A. Sloane, Table of n, a(n) for n = 1..2400

Cf. A354169, A354680, A354757, A354758, A354767, A354773, A354774, A354778, A355150, A354780.

A354793 Hamming weight of A354783(n).

Original entry on oeis.org

0, 0, 1, 1, 2, 0, 1, 1, 2, 0, 2, 2, 3, 1, 2, 0, 1, 1, 3, 1, 2, 0, 1, 1, 2, 0, 2, 2, 3, 1, 3, 1, 2, 0, 1, 1, 2, 0, 2, 2, 3, 1, 3, 1, 2, 0, 1, 1, 2, 0, 2, 2, 3, 1, 2, 0, 1, 1, 3, 1, 2, 0, 2, 2, 3, 1, 3, 1, 2, 0, 1, 1, 2, 0, 2, 2, 3, 1, 2, 0, 1, 1, 3, 1, 2, 0, 2, 2, 3, 1, 3, 1, 2, 0, 1, 1, 2, 0, 2, 2, 3, 1, 2

Offset: 1

N. J. A. Sloane, Jul 19 2022

nonn

Conjecture: This sequence appears to have a simple structure. Encode it by making the following substitutions, in this order:
Replace the initial 28 terms 0011201120223120113120112022 by S (as usual, the start is irregular), then map:
  3 1 3 -> 7
  3 1 2 -> 6
  1 2 0 1 1 2 0 2 2 -> 9
  0 1 1 -> 2
  0 2 2 -> 4
Then it appears that the encoded sequence is the concatenation of the following blocks:
  S
  79
  79(6264)^1
  79(6264)^1
  79(6264)^3
  79(6264)^3
  79(6264)^15
  79(6264)^15
  79(6264)^31
  79(6264)^31
  79(6264)^63
  79(6264)^63
  79(6264)^127
  79(6264)^127
  ...
This is probably not the most efficient encoding, but I was happy to find any one that revealed the structure.
From Michel Dekking, Jul 23 2022: (Start)
The following is another way to present the conjecture above, which shows the close connection with sequence A355150.
Conjecture: It appears that this sequence is almost a periodic sequence, with period 12. Let x:=A354789.
If n > 28, n == 5 (mod 12) is not an element of x then (written as words)
      a(n)a(n+1)...a(n+11) = 312011312022.
If n > 28, n == 5 (mod 12) is an element of x then
      a(n)a(n+1)...a(n+11) = 313120112022.
(End)

N. J. A. Sloane, Table of n, a(n) for n = 1..4900

Cf. A354169, A354757, A354783, A355150.

cp's OEIS Frontend

A354783 If the binary expansion of A354757(n) is 1 d_1 d_2 ... d_k, then the binary expansion of a(n) is c_1 c_2 ... c_k, where c_i = 1 - d_i.

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A354758 a(n) = Product_{k = ceiling(n/2)..n-1} A090252(k).

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A354780 a(n) is the bitwise OR of (the binary expansions of) b(n+1) to b(2*n), where b is A354169.

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A354781 If the binary expansion of A354780(n) is 1 d_1 d_2 ... d_k, then the binary expansion of a(n) is c_1 c_2 ... c_k, where c_i = 1 - d_i.

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A354793 Hamming weight of A354783(n).

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