A304107 Analog for squarefree numbers when n is factored in polynomial ring GF(2)[X], so that the binary expansion of n defines the corresponding (0,1)-polynomial. These are numbers n such that the said polynomial doesn't have any duplicated irreducible divisors.

1, 2, 3, 6, 7, 9, 11, 13, 14, 18, 19, 22, 23, 25, 26, 29, 31, 33, 35, 37, 38, 41, 43, 46, 47, 49, 50, 53, 55, 58, 59, 61, 62, 66, 67, 70, 71, 73, 74, 77, 79, 82, 83, 86, 87, 89, 91, 93, 94, 97, 98, 101, 103, 106, 109, 110, 111, 113, 115, 117, 118, 121, 122, 123, 127, 129, 131, 133, 134, 137, 139, 142, 143, 145, 146, 149, 154, 155, 157, 158, 159, 161

Offset: 1

Author

Antti Karttunen, May 13 2018

Keywords

• nonn

Comments

Positions of nonzeros in A091219 and A304109. Numbers n such that A091221(n) = A091222(n).

Numbers n that cannot be expressed as n = A048720(k,A000695(m)) for any k >= 0, m >= 2.

It seems that a(n) is approximately 2n for large n. See also comments in A304110.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..32769
• Index entries for sequences related to polynomials in ring GF(2)[X]

Crossrefs

Cf. A304108 (complement), A304109 (characteristic function), A304110 (least monotonic left inverse).

Cf. A000695, A014580, A048720, A091219, A091221, A091222, A304111.

Cf. also A005117.

Programs

PARI

A304109(n) = { my(fm=factor(Pol(binary(n))*Mod(1, 2))); for(k=1, #fm~, if(fm[k, 2] > 1, return(0))); (1); };
k=0; n=0; while(k<100, n++; if(A304109(n), k++; print1(n,", ")));

Formula

For n >= 1, A304110(a(n)) = n.