A292388 -id:A292388 - cp's OEIS frontend

A292389 a(n) = SumXOR_{k=1..n} A292388(k) (where SumXOR is the analog of summation under the binary XOR operation).

2, 3, 7, 2, 5, 3, 11, 2, 13, 7, 11, 5, 23, 7, 19, 2, 17, 7, 31, 5, 37, 59, 31, 3, 37, 7, 47, 5, 41, 2, 23, 37, 2, 31, 47, 2, 29, 41, 7, 61, 11, 53, 2, 43, 19, 47, 109, 41, 97, 43, 107, 37, 113, 61, 2, 59, 107, 61, 101, 59, 97, 61, 89, 31, 127, 29, 79, 2, 67

Offset: 1

Rémy Sigrist, Sep 15 2017

All terms are prime.

Conjecturally, this sequence contains infinitely many 2's.

			a(3) = A292388(1) XOR A292388(2) XOR A292388(3) = 2 XOR 1 XOR 4 = 7.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A292388.

s=0; x=0; for (n=1, 69, for (v=1, oo, if (!bittest(s,v) && isprime(bitxor(x,v)), s+=2^v; x=bitxor(x,v); print1 (x ", "); break)))

A338502 Lexicographically earliest sequence of distinct nonnegative integers such that for any n > 0, a(1) XOR ... XOR a(n) is a square (where XOR denotes the bitwise XOR operator).

Original entry on oeis.org

0, 1, 5, 4, 9, 8, 17, 16, 25, 24, 37, 21, 33, 20, 13, 45, 32, 29, 40, 48, 65, 36, 53, 72, 61, 52, 80, 57, 93, 64, 85, 49, 81, 88, 56, 96, 117, 100, 68, 125, 105, 116, 101, 120, 112, 73, 89, 137, 84, 109, 141, 180, 113, 133, 160, 132, 161, 152, 121, 144, 128

Offset: 1

Rémy Sigrist, Oct 31 2020

All terms belong to A042948.

			The first terms, alongside a(1) XOR ... XOR a(n), are:
  n   a(n)  a(1) AND ... AND a(n)
  --  ----  ---------------------
   1     0                0 = 0^2
   2     1                1 = 1^2
   3     5                4 = 2^2
   4     4                0 = 0^2
   5     9                9 = 3^2
   6     8                1 = 1^2
   7    17               16 = 4^2
   8    16                0 = 0^2
   9    25               25 = 5^2
  10    24                1 = 1^2
  11    37               36 = 6^2
  12    21               49 = 7^2

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A338502

Cf. A042948, A042964, A292388 (prime variant), A338503.

PARI
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cp's OEIS Frontend

A292389 a(n) = SumXOR_{k=1..n} A292388(k) (where SumXOR is the analog of summation under the binary XOR operation).

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