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.

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, 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, 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

Offset: 1

Author

Antti Karttunen, Jan 06 2025

Keywords

• nonn

Comments

A379505 is a variant where two 1's are considered indistinguishable. See comments there.

Examples

			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.
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:
   1_a + 1_b + 2 + 6 + 36    = 3 + 4 + 9 + 12 + 18,
   1_a + 2 + 3 + 4 + 36      = 1_b + 9 + 6 + 12 + 18,
   1_b + 2 + 3 + 4 + 36      = 1_a + 9 + 6 + 12 + 18,
   1_a + 3 + 6 + 36          = 1_b + 2 + 4 + 9 + 12 + 18,
   1_b + 3 + 6 + 36          = 1_a + 2 + 4 + 9 + 12 + 18,
   1_a + 9 + 36              = 1_b + 2 + 3 + 4 + 6 + 12 + 18,
   1_b + 9 + 36              = 1_a + 2 + 3 + 4 + 6 + 12 + 18,
   4 + 6 + 36                = 1_a + 1_b + 2 + 3 + 9 + 12 + 18,
where each sum on the left and right hand side gives (sigma(36)+1)/2 = 46.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..100000

Crossrefs

Cf. A083206, A103977, A156942, A379502, A379503 (positions of nonzero terms), A379505 (variant where two 1's are considered indistinguishable).

Cf. A000079 (conjectured to give the positions of 1's).

Programs

PARI

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)));
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);

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.

Formula

a(n) >= A379505(n).

A103977(n) = 1 <=> a(n) > 0.