A379129 a(n) is the number of unitary proper divisors d > 1 of n for which A048720(A065621(sigma(d)),sigma(n/d)) is equal to sigma(n).

0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 2, 2, 0, 1, 0, 1, 0, 1, 0, 5, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 1, 1, 2, 0, 0, 2, 1, 1, 1, 0, 3, 0, 1, 1, 0, 1, 5, 0, 0, 1, 5, 0, 1, 0, 0, 1, 1, 1, 3, 0, 1, 0, 0, 0, 3, 0, 1, 1, 1, 0, 0, 1, 1, 2, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 3

Offset: 1

Author

Antti Karttunen, Dec 18 2024

Keywords

• nonn

Examples

			For n = 21 = 3*7, both A048720(A065621(sigma(3)),sigma(7)) [= A048720(4,8)] and A048720(A065621(sigma(7)),sigma(3)) [= A048720(8,4)] yield the decided result, which is 32 = sigma(21), therefore a(21) = 2.
For n = 34 = 2*17, neither A048720(A065621(sigma(2)),sigma(17)) = A048720(7,18) = 126 nor A048720(A065621(sigma(17)),sigma(2)) = A048720(50,3) = 86 is the decided result, 54 = sigma(34), therefore a(34) = 0.
See example in A379121 why a(383942431613601) = 2.
		

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, A048720, A065621, A277320, A379113, A379114 (positions of terms > 0), A379118, A379130.

Programs

PARI

A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1,n+n-1);
A379129(n) = { my(s=sigma(n)); sumdiv(n,d,if(1==d || d==n || 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, 1A048720(A065621(sigma(d)),sigma(n/d)) == sigma(n)], where [ ] is the Iverson bracket.

a(n) <= A379130(n).