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

1,18

Comments

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

Links

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

Formula

a(n) >= A379505(n).

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

Example

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.

Prog

(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.

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).

Sequence in context: A089811 A091888 A379505 * A083928 A074038 A204843

Adjacent sequences: A379501 A379502 A379503 * A379505 A379507 A379513

Keyword

nonn,new

Author

Antti Karttunen, Jan 06 2025

Status

approved