A325559 Numbers n such that for any divisor d of n, and some integer k, A048720(d,k) = n only for trivial cases d=1 and d=n.

2, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 55, 59, 61, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 91, 97, 101, 103, 107, 109, 113, 115, 117, 121, 127, 131, 137, 139, 143, 145, 149, 151, 157, 163, 167, 169, 171, 173, 179, 181, 185, 191, 193, 197, 199, 203, 205, 209, 211, 213, 223, 227, 229, 233

Offset: 1

Author

Antti Karttunen, May 11 2019

Keywords

• nonn

Comments

These are numbers n such that there are only two divisor pairs (d, n/d) [namely, the trivial pairs (1, n) and (n, 1)] that satisfy the condition that when their binary expansions are converted to (0,1)-polynomials (e.g., 13=1101[2] encodes X^3 + X^2 + 1), then their product is the (0,1)-polynomial similarly converted from n, when the multiplication is done over field GF(2).

Differs from A206074 for the first time at n=173, where a(173) = 555, a value missing from A206074, while the first three terms of A206074 not present in this sequence are k = 689, 781 and 913, for all of which A325560(k) = 3, not 2.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..25000
• Index entries for sequences related to polynomials in ring GF(2)[X]

Crossrefs

Cf. A048720, A206074.

Positions of 2's in A325560, positions of 1's in A325563 (after the initial 1), fixed points of A325643 (after the initial 1).

Some subsequences: A257688 (after its initial 1), A325386 (the remaining terms).

Programs

PARI

A325560(n) = { my(p = Pol(binary(n))*Mod(1, 2)); sumdiv(n,d,my(q = Pol(binary(d))*Mod(1, 2)); !(p%q)); };
isA325559(n) = (2 == A325560(n));