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%I #26 May 28 2019 19:33:31
%S 1,1,1,3,1,2,1,4,3,3,1,13,1,2,2,15,1,15,1,9,4,3,1,33,3,2,4,12,1,40,1,
%T 18,2,3,4,201,1,2,4,33,1,40,1,9,7,3,1,245,3,20,2,15,1,25,4,34,4,3,1,
%U 577,1,2,15,63,4,40,1,9,2,44,1,951,1,2,15,10,4,34,1,164,15,3,1,864,4,2,2,34,1,592,2,9,4,3,2,577,1,21,7,210,1,40,1,29,40
%N Number of ways to partition the divisors of n into two complementary sets whose sums are relatively prime. (Here only distinct unordered pairs of such subsets are counted.)
%C a(n) is the number of such subsets of divisors of n that include {1} and have sum that is coprime to the sum of their complement.
%C Records 1, 3, 4, 13, 15, 33, 40, 201, 245, 577, 951, 8672, 14595, 33904, 168904, 253694, 2057413, 2395584, 2396158, 2571028, 159504796, 572644864, ... occur at positions 1, 4, 8, 12, 16, 24, 30, 36, 48, 60, 72, 120, 144, 180, 240, 336, 360, 420, 480, 630, 720, 840, ...
%H Antti Karttunen, <a href="/A325806/b325806.txt">Table of n, a(n) for n = 1..1079</a>
%F a(n) <= A100577(n).
%e For n = 1, its divisor set [1] can be partitioned only to an empty set [] and set [1], with sums 0 and 1 respectively, and gcd(0,1) = 1, thus this partitioning is included, and a(1) = 1.
%e For n = 3, its divisor set [1, 3] can be partitioned as [] and [1,3] (sums 0 and 4), [1] and [3] (sums 1 and 3), and only in latter case the sums are coprime as gcd(1,3) = 1, thus a(3) = 1, and similarly a(p) = 1 for any other prime p as well.
%e For n = 6, its divisor set [1, 2, 3, 6] can be partitioned as [] and [1, 2, 3, 6] (sums 0 and 12), [1, 2] and [3, 6] (sums 3 and 9), [1, 3] and [2, 6] (sums 4 and 8), [2] and [1, 3, 6] (sums 2 and 10), [3] and [1, 2, 6] (sums 3 and 9), [6] and [1, 2, 3] (sums 6 and 6), and also as [1] and [2, 3, 6] (sums 1 and 11), and [1, 6] and [2, 3] (sums 7 and 5) and only in latter two cases their sums are coprime, thus a(6) = 2.
%e For n = 12, its divisor set [1, 2, 3, 4, 6, 12] can be partitioned altogether in 2^(6-1) = 32 ways, but of which only the following thirteen partitions have coprime sums:
%e [1] and [2, 3, 4, 6, 12],
%e [1, 2] and [3, 4, 6, 12],
%e [1, 4] and [2, 3, 6, 12],
%e [1, 2, 6] and [3, 4, 12],
%e [1, 4, 6] and [2, 3, 12],
%e [1, 2, 4, 6] and [3, 12],
%e [1, 12] and [2, 3, 4, 6],
%e [1, 2, 12] and [3, 4, 6],
%e [1, 4, 12] and [2, 3, 6],
%e [1, 2, 4, 12] and [3, 6],
%e [1, 6, 12] and [2, 3, 4],
%e [1, 4, 6, 12] and [2, 3],
%e [1, 2, 4, 6, 12] and [3],
%e thus a(12) = 13.
%t Array[Function[d, Count[DeleteDuplicates[Sort /@ Map[{#, Complement[d, #]} &, Subsets@ d]], _?(CoprimeQ @@ (Total /@ #) &)]]@ Divisors@ # &, 105] (* _Michael De Vlieger_, May 27 2019 *)
%o (PARI)
%o A325806(n) = { my(divs=divisors(n), s=sigma(n)); sum(b=0,(2^(-1+length(divs)))-1,(1==gcd(s,sumbybits(divs,2*b)))); };
%o sumbybits(v,b) = { my(s=0,i=1); while(b>0,s += (b%2)*v[i]; i++; b >>= 1); (s); };
%Y Cf. A000005, A000203, A100577, A325807.
%Y Cf. A083206, A083207, A083209, A083210.
%K nonn
%O 1,4
%A _Antti Karttunen_, May 24 2019
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