A379130 a(n) is the number of unitary divisors d of n for which A048720(A065621(sigma(d)),sigma(n/d)) is equal to sigma(n).

1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 4, 3, 1, 2, 1, 2, 1, 2, 1, 6, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 2, 2, 3, 1, 1, 3, 2, 2, 2, 1, 4, 1, 2, 2, 1, 2, 6, 1, 1, 2, 6, 1, 2, 1, 1, 2, 2, 2, 4, 1, 2, 1, 1, 1, 4, 1, 2, 2, 2, 1, 1, 2, 2, 4, 3, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 4

Offset: 1

Author

Antti Karttunen, Dec 18 2024

Keywords

• nonn

Comments

It seems that A046528 gives all numbers k for which a(k) = A034444(k).

Examples

			For every n, a(n) >= 1, because A048720(A065621(sigma(1)), sigma(n)) = A048720(A065621(1), sigma(n)) = A048720(1, sigma(n)) = sigma(n).
For n = 21 = 3*7, after the divisor pair [1,21], all other divisor pairs also satisfy the condition: A048720(A065621(sigma(3)),sigma(7)) [= A048720(4,8)] and A048720(A065621(sigma(7)),sigma(3)) [= A048720(8,4)] and A048720(A065621(sigma(21)),sigma(1)) [= A048720(32,1)] all yield the decided result, 32 = sigma(21), therefore a(21) = 4.
See also examples in A379129.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences defined by congruent products between domains N and GF(2)[X].
• Index entries for sequences related to polynomials in ring GF(2)[X].
• Index entries for sequences related to sigma(n).

Crossrefs

Cf. A000203, A034444, A046528, A048720, A065621, A277320, A379113, A379129.

Cf. also A325565.

Programs

PARI

A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1,n+n-1);
A379130(n) = { my(s=sigma(n)); sumdiv(n,d,if(1!=gcd(d,n/d), 0, A048720(A065621(sigma(n/d)),sigma(d))==s)); };

Formula

a(n) = Sum_{d|n, gcd(d,n/d)=1} [A048720(A065621(sigma(d)),sigma(n/d)) == sigma(n)], where [ ] is the Iverson bracket.

A379129(n) <= a(n) <= A034444(n).