A335019 Numbers whose prime exponents generates rotationally symmetric XOR-triangles.

1, 2, 4, 8, 16, 32, 42, 64, 70, 128, 256, 390, 512, 1024, 1122, 1764, 2002, 2048, 2210, 4096, 4900, 6270, 7854, 8192, 9450, 15470, 15750, 16384, 26460, 32768, 36366, 54978, 58786, 65536, 66990, 73500, 74088, 86710, 108290, 123480, 131072, 152100, 162690

Offset

1,2

Comments

For any positive number with prime factorization Product_{k = 1..m} prime(k)^e_k (where prime(k) denotes the k-th prime number and e_m > 0), we build a XOR-triangle with (e_m, ..., e_1) as top row, and having each entry in the subsequent rows be the XOR of the two values above it. This sequence lists integers whose XOR-triangle has 3-fold rotational symmetry. A334990 gives the second row of such XOR-triangles for numbers that are not powers of 2.

This sequence has strong connections with A334556: for any n > 0 and k > 0, A019565(A334556(n))^k belongs to this sequence.

Every power of 2 belongs to the sequence.

If m belongs to this sequence, then m^2 also belongs to this sequence.

Links

Table of n, a(n) for n=1..43.

Index entries for sequences related to XOR-triangles

Formula

A067029(a(n)) = A071178(a(n)).

Example

The number 15750 = 7^1 * 5^2 * 3^3 * 2^1 yields the following XOR-triangle:
     1 2 3 1
      3 1 2
       2 3
        1
As this XOR-triangle has rotational symmetry, 15750 belongs to this sequence.

Prog

(PARI) is(n) = {
    my (e);
    if (n==1, e=[],
        my (f=factor(n), m=primepi(f[#f~, 1]));
        e=vector(m, k, valuation(n, prime(m+1-k)))
    );
    my (x=e);
    for (k=1, #e,
        if (e[k]!=x[#x], return (0));
        if (x[1]!=e[#e+1-k], return (0));
        x=vector(#x-1, k, bitxor(x[k], x[k+1]));
    );
    return (1);
}

Crossrefs

Cf. A019565, A067029, A071178, A334556, A334990.

Sequence in context: A353720 A371294 A133408 * A337719 A289275 A316758

Adjacent sequences: A335016 A335017 A335018 * A335020 A335021 A335022

Keyword

nonn,base

Author

Rémy Sigrist, May 19 2020

Status

approved