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

1,10

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, 1<d<n} [A048720(A065621(sigma(d)),sigma(n/d)) == sigma(n)], where [ ] is the Iverson bracket.

a(n) <= A379130(n).

Example

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.

Prog

(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)); };

Crossrefs

Cf. A000203, A048720, A065621, A277320, A379113, A379114 (positions of terms > 0), A379118, A379130.

Sequence in context: A341979 A331838 A361017 * A227836 A364042 A033764

Adjacent sequences: A379126 A379127 A379128 * A379130 A379131 A379132

Keyword

nonn

Author

Antti Karttunen, Dec 18 2024

Status

approved