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

1,3

Comments

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

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).

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).

Example

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.

Prog

(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)); };

Crossrefs

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

Cf. also A325565.

Sequence in context: A141591 A174545 A102523 * A323023 A083415 A115514

Adjacent sequences: A379127 A379128 A379129 * A379131 A379132 A379133

Keyword

nonn

Author

Antti Karttunen, Dec 18 2024

Status

approved