cp's OEIS Frontend

A379504 a(n) is the number of ways of partitioning the divisors of n into two disjoint sets with equal sum, when an extra 1-divisor is added to the divisor set, and the two 1-divisors are considered distinct from each other.

%I A379504 #34 Jan 07 2025 15:56:05
%S A379504 1,1,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,
%T A379504 0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,
%U A379504 0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1
%N A379504 a(n) is the number of ways of partitioning the divisors of n into two disjoint sets with equal sum, when an extra 1-divisor is added to the divisor set, and the two 1-divisors are considered distinct from each other.
%C A379504 A379505 is a variant where two 1's are considered indistinguishable. See comments there.
%H A379504 Antti Karttunen, <a href="/A379504/b379504.txt">Table of n, a(n) for n = 1..100000</a>
%F A379504 a(n) >= A379505(n).
%F A379504 A103977(n) = 1 <=> a(n) > 0.
%e A379504 a(18) = 2 as its divisor set with an extra 1 is [1_a, 1_b, 2, 3, 6, 9, 18], and this can be partitioned to two sets with equal sums either as 1_a+1_b+3+6+9 = 2+18 or as 2+3+6+9 = 1_a+1_b+18.
%e A379504 a(36) = 8 as its divisor set with an extra 1 is [1_a, 1_b, 2, 3, 4, 6, 9, 12, 18, 36], and this can be partitioned in any of the following ways:
%e A379504    1_a + 1_b + 2 + 6 + 36    = 3 + 4 + 9 + 12 + 18,
%e A379504    1_a + 2 + 3 + 4 + 36      = 1_b + 9 + 6 + 12 + 18,
%e A379504    1_b + 2 + 3 + 4 + 36      = 1_a + 9 + 6 + 12 + 18,
%e A379504    1_a + 3 + 6 + 36          = 1_b + 2 + 4 + 9 + 12 + 18,
%e A379504    1_b + 3 + 6 + 36          = 1_a + 2 + 4 + 9 + 12 + 18,
%e A379504    1_a + 9 + 36              = 1_b + 2 + 3 + 4 + 6 + 12 + 18,
%e A379504    1_b + 9 + 36              = 1_a + 2 + 3 + 4 + 6 + 12 + 18,
%e A379504    4 + 6 + 36                = 1_a + 1_b + 2 + 3 + 9 + 12 + 18,
%e A379504 where each sum on the left and right hand side gives (sigma(36)+1)/2 = 46.
%o A379504 (PARI)
%o A379504 partitions_into_distinct_parts(n, parts, from=1) = if(!n, 1, if(from>#parts, 0, my(s=0); for(i=from, #parts, if(parts[i]<=n, s += partitions_into_distinct_parts(n-parts[i], parts, i+1))); (s)));
%o A379504 A379504(n) = if(!issquare(n) && !issquare(2*n), 0, my(divs=concat(1,divisors(n)), s=sigma(n)); partitions_into_distinct_parts((s+1)/2, vecsort(divs,,4))/2);
%o A379504 (PARI) A379504(n) = if(!issquare(n) && !issquare(2*n), 0, my(p=('x^1 + 'x^-1)); fordiv(n, d, p *= ('x^d + 'x^-d)); (polcoeff(p, 0)/2)); \\ Faster program, after code in A083206.
%Y A379504 Cf. A083206, A103977, A156942, A379502, A379503 (positions of nonzero terms), A379505 (variant where two 1's are considered indistinguishable).
%Y A379504 Cf. A000079 (conjectured to give the positions of 1's).
%K A379504 nonn
%O A379504 1,18
%A A379504 _Antti Karttunen_, Jan 06 2025