A265885 -id:A265885 - cp's OEIS frontend

A265705 Triangle read by rows: T(n,k) = k IMPL n, 0 <= k <= n, bitwise logical IMPL.

0, 1, 1, 3, 2, 3, 3, 3, 3, 3, 7, 6, 5, 4, 7, 7, 7, 5, 5, 7, 7, 7, 6, 7, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 15, 14, 13, 12, 11, 10, 9, 8, 15, 15, 15, 13, 13, 11, 11, 9, 9, 15, 15, 15, 14, 15, 14, 11, 10, 11, 10, 15, 14, 15, 15, 15, 15, 15, 11, 11, 11, 11, 15

Offset: 0

Reinhard Zumkeller, Dec 15 2015

			.          10 | 1010                            12 | 1100
.           4 |  100                             6 |  110
.   ----------+-----                     ----------+-----
.   4 IMPL 10 | 1011 -> T(10,4)=11       6 IMPL 12 | 1101 -> T(12,6)=13
.
First 16 rows of the triangle, where non-symmetrical rows are marked, see comment concerning A158582 and A089633:
.   0:                                 0
.   1:                               1   1
.   2:                             3   2   3
.   3:                           3   3   3   3
.   4:                         7   6   5   4   7    X
.   5:                       7   7   5   5   7   7
.   6:                     7   6   7   6   7   6   7
.   7:                   7   7   7   7   7   7   7   7
.   8:                15  14  13  12  11  10   9   8  15    X
.   9:              15  15  13  13  11  11   9   9  15  15    X
.  10:            15  14  15  14  11  10  11  10  15  14  15    X
.  11:          15  15  15  15  11  11  11  11  15  15  15  15
.  12:        15  14  13  12  15  14  13  12  15  14  13  12  15    X
.  13:      15  15  13  13  15  15  13  13  15  15  13  13  15  15
.  14:    15  14  15  14  15  14  15  14  15  14  15  14  15  14  15
.  15:  15  15  15  15  15  15  15  15  15  15  15  15  15  15  15  15 .

Reinhard Zumkeller, Rows n = 0..255 of triangle, flattened
Eric Weisstein's World of Mathematics, Implies

Cf. A003817, A007088, A029578, A089633, A158582, A247648, A265716 (central terms), A265736 (row sums).
Cf. A053644, A265885, A327490.
Other triangles: A080099 (AND), A080098 (OR), A051933 (XOR), A102037 (CNIMPL).

a265705_tabl = map a265705_row [0..]
a265705_row n = map (a265705 n) [0..n]
a265705 n k = k `bimpl` 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
for n in 0:15 println(n == 0 ? [0] : [Bits("IMP", k, n) for k in 0:n]) end  # Peter Luschny, Sep 25 2021

A265705 := (n, k) -> Bits:-Implies(k, n):
seq(seq(A265705(n, k), k=0..n), n=0..11); # Peter Luschny, Sep 23 2019

T[n_, k_] := If[n == 0, 0, BitOr[2^Length[IntegerDigits[n, 2]]-1-k, n]];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 25 2021, after David A. Corneth's PARI code *)

T(n, k) = if(n==0,return(0)); bitor((2<David A. Corneth, Sep 24 2021

T(n,0) = T(n,n) = A003817(n).
T(2*n,n) = A265716(n).
Let m = A089633(n): T(m,k) = T(m,m-k), k = 0..m.
Let m = A158582(n): T(m,k) != T(m,m-k) for at least one k <= n.
Let m = A247648(n): T(2*m,m) = 2*m.
For n > 0: A029578(n+2) = number of odd terms in row n; no even terms in odd-indexed rows.
A265885(n) = T(prime(n),n).
A053644(n) = smallest k such that row k contains n.

A070883 Bitwise XOR of n and n-th prime.

Original entry on oeis.org

3, 1, 6, 3, 14, 11, 22, 27, 30, 23, 20, 41, 36, 37, 32, 37, 42, 47, 80, 83, 92, 89, 68, 65, 120, 127, 124, 119, 112, 111, 96, 163, 168, 169, 182, 179, 184, 133, 128, 133, 154, 159, 148, 237, 232, 233, 252, 239, 210, 215, 218, 219, 196, 205, 310, 319, 308, 309, 302

Offset: 1

Reinhard Zumkeller, May 22 2002

For any integer k, XOR(n,k) = 2*OR(n,k) - (n+k). - Gary Detlefs, Oct 26 2013

			A000040(25)=97, [25]2 = '00011001', [97]2 = '01100001' '00011001' XOR '01100001' = '01111000', therefore a(25)=120.

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

Cf. A169810, A169811, A194187.

Cf. A265885 (IMPL).

import Data.Bits (xor)
a070883 n = a070883_list !! (n-1)
a070883_list = zipWith xor [1..] a000040_list
-- Reinhard Zumkeller, Jun 23 2015

Table[ BitXor[ n, Prime[n]], {n, 1, 55}]

PARI
```
a(n) = bitxor(n, prime(n));
```

from sympy import prime
def a(n): return n^prime(n)
print([a(n) for n in range(1, 60)]) # Michael S. Branicky, Mar 05 2022

a(n) = 2*OR(p,n) - (p+n), for n-th prime p. - Gary Detlefs, Oct 26 2013

cp's OEIS Frontend

A265705 Triangle read by rows: T(n,k) = k IMPL n, 0 <= k <= n, bitwise logical IMPL.

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