A391571 a(n) is the value of A280500(n*k, k) for the smallest odd k > 1 for which this value is not zero, where A280500 implements the carryless base-2 division, and returns 0 if the division leaves nonzero remainder.

1, 2, 7, 4, 5, 14, 11, 8, 9, 10, 31, 28, 29, 22, 27, 16, 17, 18, 23, 20, 21, 62, 59, 56, 57, 58, 43, 44, 37, 54, 35, 32, 33, 34, 39, 36, 37, 46, 63, 40, 41, 42, 127, 124, 125, 118, 123, 112, 113, 114, 119, 116, 117, 86, 99, 88, 89, 74, 111, 108, 109, 70, 107, 64, 65, 66, 71, 68, 69, 78, 75, 72, 73, 74, 95, 92, 93, 126, 91

Offset

1,2

Comments

Search for the smallest odd k > 1 such that k*n = A048720(k,m) for some m, and then set a(n) = m. A391570(n) gives the k.

The smallest term k of A004780 for which a(k) = k is A004780(940) = 1117. See A391581.

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences defined by congruent products between domains N and GF(2)[X].

Index entries for sequences related to binary expansion of n.

Index entries for sequences related to carryless arithmetic.

Formula

A391570(n) * n = A048720(A391570(n), a(n)).

a(n) >= n. [Implied by above, as x*y >= A048720(x,y)]

a(A003714(n)) = A003714(n), and a(A391581(n)) = A391581(n).

Prog

(PARI) A391571(n) = forstep(k=3, oo, 2, my(Pnk=Pol(binary(n*k))*Mod(1, 2), Pk=Pol(binary(k))*Mod(1, 2)); if(0==lift(Pnk % Pk), return(fromdigits(Vec(lift(Pnk / Pk)), 2))));

Crossrefs

Cf. A048720, A280500, A391570, A391580 (terms k for which a(k) > k).

Column 2 of A391726 (cf. also A391583, which is column 3. Compare also their scatter plots).

Fixed points: (A003714 U A391581).

Cf. also A115857, A391572, A391573.

Sequence in context: A325644 A198036 A316249 * A197143 A132724 A306579

Adjacent sequences: A391568 A391569 A391570 * A391572 A391573 A391574

Keyword

nonn,base

Author

Antti Karttunen, Dec 15 2025

Status

approved