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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).
2
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
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
KEYWORD
nonn,new
AUTHOR
Antti Karttunen, Dec 18 2024
STATUS
approved