A004443 -id:A004443 - cp's OEIS frontend

A359946 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, n XOR a(n) is a prime number (where XOR denotes the bitwise XOR operator).

2, 1, 4, 3, 6, 5, 10, 11, 12, 7, 8, 9, 14, 13, 16, 15, 18, 17, 20, 19, 22, 21, 26, 27, 28, 23, 24, 25, 30, 29, 34, 35, 36, 31, 32, 33, 38, 37, 42, 43, 44, 39, 40, 41, 46, 45, 48, 47, 50, 49, 52, 51, 54, 53, 58, 59, 60, 55, 56, 57, 62, 61, 64, 63, 66, 65, 68

Offset: 1

Rémy Sigrist, Jan 19 2023

This sequence is a self-inverse permutation of the positive integers.
See A359947 for the corresponding prime numbers.
When considering nonnegative integers instead of positive integers, we obtain A004443.

			The first terms, alongside n XOR a(n), are:
  n   a(n)  n XOR a(n)
  --  ----  ----------
   1     2           3
   2     1           3
   3     4           7
   4     3           7
   5     6           3
   6     5           3
   7    10          13
   8    11           3
   9    12           5
  10     7          13
  11     8           3
  12     9           5

Index entries for sequences that are permutations of the natural numbers

Cf. A004443, A359947.

nn = 2^10; c[] := False; a[1] = 2; c[2] = True; u = 1; Do[k = u; While[Nand[! c[k], PrimeQ@ BitXor[n, k]], k++]; Set[{a[n], c[k]}, {k, True}]; If[k == u, While[c[u], u++]], {n, 2, nn}]; Array[a, nn] (* _Michael De Vlieger, Jan 21 2023 *)

{ s = 0; for (n=1, 67, for (v=1, oo, if (!bittest(s, v) && isprime(bitxor(n, v)), print1 (v", "); s += 2^v; break))) }

A352526 a(n) = Product_{k=0..n} Nimsum (2*k + 2), with Nimsum (2 + 2) = 0 replaced by 1.

Original entry on oeis.org

2, 2, 12, 48, 480, 3840, 53760, 645120, 11612160, 185794560, 4087480320, 81749606400, 2125489766400, 51011754393600, 1530352631808000, 42849873690624000, 1456895705481216000, 46620662575398912000, 1771585177865158656000, 63777066403145711616000, 2678636788932119887872000

Offset: 0

Peter McNair, Mar 19 2022

nonn

Nimsum 2*k + 2 = A004443(2*k).
Sum_{n>0} 1/a(n) = 1/sqrt(e) = A092605.
Sum_{n>0} 1/a(2*n-1) = sinh(1/2) = A334367.
Sum_{n>0} 1/a(2*n) = cosh(1/2) - 2*sinh(1/2).
a(n)/2^n = abs(A265376(n+1)) = Product_{k=0..n} Nimsum k + 1, with Nimsum 1 + 1 = 0 replaced by 1, n > 0.

Cf. A004443, A092605, A265376, A334367.

a[n_] := Product[If[k == 1, 1, BitXor[2*k, 2]], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Mar 19 2022 *)

PARI
```
a(n) = 2*prod(k=2,n,bitxor(2*k, 2))
```

a(n) = 2*Product_{k=2..n} A004443(2*k).
a(n) = 2^(n-1)*(n+1)!/floor((n+1)/2), n > 0.
a(n) = 2^(n-1)*(1+(-1)^n)*((n-1)!+n!)-((-1)^n-1)*(2*n)!!/2, n > 0.
a(n) = 2*a(n-1)*(n+(-1)^n), n > 1, with a(1) = 2.

cp's OEIS Frontend

A359946 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, n XOR a(n) is a prime number (where XOR denotes the bitwise XOR operator).

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