A379129 - OEIS

%I #20 Dec 18 2024 15:05:55

%S 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,

%T 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,

%U 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

%N 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).

%H Antti Karttunen, <a href="/A379129/b379129.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Con#CongruCrossDomain">Index entries for sequences defined by congruent products between domains N and GF(2)[X]</a>.

%H <a href="/index/Ge#GF2X">Index entries for sequences related to polynomials in ring GF(2)[X]</a>.

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.

%F 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.

%F a(n) <= A379130(n).

%e 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.

%e 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.

%e See example in A379121 why a(383942431613601) = 2.

%o (PARI)

%o A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);

%o A065621(n) = bitxor(n-1,n+n-1);

%o 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)); };

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

%K nonn,new

%O 1,10

%A _Antti Karttunen_, Dec 18 2024