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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).

%I A379130 #27 Dec 18 2024 15:05:30
%S A379130 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,
%T A379130 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,
%U A379130 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
%N 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).
%C A379130 It seems that A046528 gives all numbers k for which a(k) = A034444(k).
%H A379130 Antti Karttunen, <a href="/A379130/b379130.txt">Table of n, a(n) for n = 1..65537</a>
%H A379130 <a href="/index/Con#CongruCrossDomain">Index entries for sequences defined by congruent products between domains N and GF(2)[X]</a>.
%H A379130 <a href="/index/Ge#GF2X">Index entries for sequences related to polynomials in ring GF(2)[X]</a>.
%H A379130 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A379130 a(n) = Sum_{d|n, gcd(d,n/d)=1} [A048720(A065621(sigma(d)),sigma(n/d)) == sigma(n)], where [ ] is the Iverson bracket.
%F A379130 A379129(n) <= a(n) <= A034444(n).
%e A379130 For every n, a(n) >= 1, because A048720(A065621(sigma(1)), sigma(n)) = A048720(A065621(1), sigma(n)) = A048720(1, sigma(n)) = sigma(n).
%e A379130 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.
%e A379130 See also examples in A379129.
%o A379130 (PARI)
%o A379130 A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
%o A379130 A065621(n) = bitxor(n-1,n+n-1);
%o A379130 A379130(n) = { my(s=sigma(n)); sumdiv(n,d,if(1!=gcd(d,n/d), 0, A048720(A065621(sigma(n/d)),sigma(d))==s)); };
%Y A379130 Cf. A000203, A034444, A046528, A048720, A065621, A277320, A379113, A379129.
%Y A379130 Cf. also A325565.
%K A379130 nonn
%O A379130 1,3
%A A379130 _Antti Karttunen_, Dec 18 2024