A265705 -id:A265705 - cp's OEIS frontend

A102037 Triangle of BitAnd(BitNot(n), k).

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 3, 0, 0, 0, 2, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 0, 0, 0, 2, 2, 4, 4, 6, 6, 0, 0, 0, 1, 0, 1, 4, 5, 4, 5, 0, 1, 0, 0, 0, 0, 0, 4, 4, 4, 4, 0, 0, 0, 0, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0

Offset: 0

Eric W. Weisstein, Dec 25 2004

As a logical operation on two variables this is also called the 'converse nonimplication'. - Peter Luschny, Sep 25 2021

			Table starts:
[0] 0;
[1] 0, 0;
[2] 0, 1, 0;
[3] 0, 0, 0, 0;
[4] 0, 1, 2, 3, 0;
[5] 0, 0, 2, 2, 0, 0;
[6] 0, 1, 0, 1, 0, 1, 0;
[7] 0, 0, 0, 0, 0, 0, 0, 0;
[8] 0, 1, 2, 3, 4, 5, 6, 7, 0;
[9] 0, 0, 2, 2, 4, 4, 6, 6, 0, 0.

Eric Weisstein's World of Mathematics, Sierpinski Sieve
Wikipedia, Converse nonimplication.

Cf. A350094 (row sums), A268040 (array).

Other triangles: A080099 (AND), A080098 (OR), A051933 (XOR), A265705 (IMPL).

using IntegerSequences
A102037Row(n) = [Bits("CNIMP", n, k) for k in 0:n]
for n in 0:20 println(A102037Row(n)) end  # Peter Luschny, Sep 25 2021

with(Bits): cnimp := (n, k) -> And(Not(n), k):
seq(print(seq(cnimp(n,k), k=0..n)), n = 0..12); # Peter Luschny, Sep 25 2021

A265716 a(n) = n IMPL (2*n), where IMPL is the bitwise logical implication.

Original entry on oeis.org

0, 2, 5, 6, 11, 10, 13, 14, 23, 22, 21, 22, 27, 26, 29, 30, 47, 46, 45, 46, 43, 42, 45, 46, 55, 54, 53, 54, 59, 58, 61, 62, 95, 94, 93, 94, 91, 90, 93, 94, 87, 86, 85, 86, 91, 90, 93, 94, 111, 110, 109, 110, 107, 106, 109, 110, 119, 118, 117, 118, 123, 122

Offset: 0

Reinhard Zumkeller, Dec 15 2015

The scatterplot exhibits fractal qualities. - Bill McEachen, Dec 27 2022

			.      2*21=42 | 101010                      2*6=12 | 1100
.           21 |  10101                           6 |  110
.   -----------+-------                   ----------+-----
.   21 IMPL 42 | 101010 -> a(21) = 42     6 IMPL 12 | 1101 -> a(6) = 13 .

Reinhard Zumkeller, Table of n, a(n) for n = 0..8191 <= 2^13-1
Eric Weisstein's World of Mathematics, Implies

Cf. A265705, A247648, A003817.

a265716 n = n `bimpl` (2 * n) where
   bimpl 0 0 = 0
   bimpl p q = 2 * bimpl p' q' + if u <= v then 1 else 0
               where (p', u) = divMod p 2; (q', v) = divMod q 2

A265716 := n -> Bits:-Implies(n, 2*n):
seq(A265716(n), n=0..61); # Peter Luschny, Sep 23 2019

IMPL[n_, k_] := If[n == 0, 0, BitOr[2^Length[IntegerDigits[k, 2]]-1-n, k]];
a[n_] := n ~IMPL~ (2n);
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 16 2021 *)

a(n)=bitor(bitneg(n, exponent(n)+1), 2*n) \\ Charles R Greathouse IV, Jan 20 2023

a(n) = A265705(2*n,n): central terms of triangle A265705;
a(A247648(n)) = 2*A247648(n).
a(n)= bitor(A003817(n)-n, 2*n) (conjectured). - Bill McEachen, Dec 13 2021
2n <= a(n) <= 3n. - Charles R Greathouse IV, Jan 20 2023

A265885 a(n) = n IMPL prime(n), where IMPL is the bitwise logical implication.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 25, 23, 23, 29, 31, 55, 59, 59, 63, 63, 63, 61, 111, 111, 107, 111, 123, 127, 103, 101, 103, 107, 111, 113, 127, 223, 223, 223, 221, 223, 223, 251, 255, 255, 247, 245, 255, 211, 215, 215, 211, 223, 239, 237, 237, 239, 251, 251, 457, 455

Offset: 1

Reinhard Zumkeller, Dec 17 2015

nonn

			.   prime(25)=97 | 1100001
.             25 |   11001
.   -------------+--------
.     25 IMPL 97 | 1100111 -> a(25) = 103 .

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Implies

Cf. A000040, A004676, A007088, A070883 (XOR), A265705.

a265885 n = n `bimpl` a000040 n where
   bimpl 0 0 = 0
   bimpl p q = 2 * bimpl p' q' + if u <= v then 1 else 0
               where (p', u) = divMod p 2; (q', v) = divMod q 2

using IntegerSequences
[Bits("IMP", n, p) for (n, p) in enumerate(Primes(1, 263))] |> println  # Peter Luschny, Sep 25 2021

a:= n-> Bits[Implies](n, ithprime(n)):
seq(a(n), n=1..56);  # Alois P. Heinz, Sep 24 2021

IMPL[n_, k_] := If[n == 0, 0, BitOr[2^Length[IntegerDigits[k, 2]]-1-n, k]];
a[n_] := n ~IMPL~ Prime[n];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Sep 25 2021, after David A. Corneth's code in A265705 *)

a(n) = bitor((2<Michel Marcus, Jan 22 2022

a(n) = A265705(A000040(n),n).

cp's OEIS Frontend

A102037 Triangle of BitAnd(BitNot(n), k).

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A265716 a(n) = n IMPL (2*n), where IMPL is the bitwise logical implication.

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A265885 a(n) = n IMPL prime(n), where IMPL is the bitwise logical implication.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Julia

Maple

Mathematica

PARI

Formula