A379505 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 indistinguishable.

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, 5, 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, 21, 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

Conjecture 1: For all n >= 1, a(A156942(n)) > 0. Also, if a(A156942(n)) > 1 is true for all n, it would imply that there are no quasiperfect numbers, numbers x with sigma(x) = 2x+1, as such numbers must all reside in A156942 and have a(x) = 1. (See references in A336700).

Conjecture 2: a(n) = 1 if and only if n = 2^k, with k >= 0. This claim is equal to to the statement that there are neither quasiperfect numbers nor almost perfect (least deficient) numbers, numbers x with sigma(x) = 2x-1, others than those given by A000079.

Links

Antti Karttunen, Table of n, a(n) for n = 1..1799 [a larger b-file requested]

Formula

a(n) <= A379504(n).

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

Example

a(18) = 2 as its divisor set with an extra 1 is [1, 1, 2, 3, 6, 9, 18], and this can be partitioned to two sets with equal sums either as 1+1+3+6+9 = 2+18 or as 2+3+6+9 = 1+1+18.
a(36) = 5 as its divisor set with an extra 1 is [1, 1, 2, 3, 4, 6, 9, 12, 18, 36], and this can be partitioned in any of the following five ways, when two 1's are considered indistinguishable:
  1+1+2+6+36  = 3+4+9+12+18,
  1+2+3+4+36  = 1+6+9+12+18,
  1+3+6+36    = 1+2+4+9+12+18,
  1+9+36      = 1+2+3+4+6+12+18,
  4+6+36      = 1+1+2+3+9+12+18,
where each sum on the left and right hand side gives (sigma(36)+1)/2 = 46.
There are 42 partitions of (sigma(72)+1)/2 = 98 into the divisors of 72, [1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72], with an extra 1 allowed:
  [2, 24, 72],
  [1, 1, 24, 72],
  [8, 18, 72],
  [2, 6, 18, 72],
  [1, 1, 6, 18, 72],
  [1, 3, 4, 18, 72],
  [1, 1, 2, 4, 18, 72],
  [1, 4, 9, 12, 72],
  [2, 3, 9, 12, 72],
  [1, 1, 3, 9, 12, 72],
  [6, 8, 12, 72],
  [2, 4, 8, 12, 72],
  [1, 1, 4, 8, 12, 72],
  [1, 2, 3, 8, 12, 72],
  [1, 3, 4, 6, 12, 72],
  [1, 1, 2, 4, 6, 12, 72],
  [3, 6, 8, 9, 72],
  [1, 2, 6, 8, 9, 72],
  [2, 3, 4, 8, 9, 72],
  [1, 1, 3, 4, 8, 9, 72],
  [1, 1, 2, 3, 4, 6, 9, 72],
  [8, 12, 18, 24, 36],
  [2, 6, 12, 18, 24, 36],
  [1, 1, 6, 12, 18, 24, 36],
  [1, 3, 4, 12, 18, 24, 36],
  [1, 1, 2, 4, 12, 18, 24, 36],
  [3, 8, 9, 18, 24, 36],
  [1, 2, 8, 9, 18, 24, 36],
  [1, 4, 6, 9, 18, 24, 36],
  [2, 3, 6, 9, 18, 24, 36],
  [1, 1, 3, 6, 9, 18, 24, 36],
  [1, 1, 2, 3, 4, 9, 18, 24, 36],
  [2, 4, 6, 8, 18, 24, 36],
  [1, 1, 4, 6, 8, 18, 24, 36],
  [1, 2, 3, 6, 8, 18, 24, 36],
  [3, 6, 8, 9, 12, 24, 36],
  [1, 2, 6, 8, 9, 12, 24, 36],
  [2, 3, 4, 8, 9, 12, 24, 36],
  [1, 1, 3, 4, 8, 9, 12, 24, 36],
  [1, 1, 2, 3, 4, 6, 9, 12, 24, 36],
  [2, 3, 4, 6, 8, 9, 12, 18, 36],
  [1, 1, 3, 4, 6, 8, 9, 12, 18, 36],
therefore a(72) = 42/2 = 21.

Prog

(PARI)
partitions_into_distinct_parts_with_extra1allowed(n, parts, from=1) = if(n<=1, 1, if(from>#parts, 0, my(s=0); for(i=from, #parts, if(parts[i]<=n, s += partitions_into_distinct_parts_with_extra1allowed(n-parts[i], parts, i+1))); (s)));
A379505(n) = if(1==n, n, if(!issquare(n) && !issquare(2*n), 0, my(divs=divisors(n), s=sigma(n)); partitions_into_distinct_parts_with_extra1allowed((s+1)/2, vecsort(divs, , 4))/2));

Crossrefs

Cf. A083206, A103977, A156942, A336700, A379502, A379503 (positions of nonzero terms), A379504 (version where two 1's are considered distinct).

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

Sequence in context: A349436 A089811 A091888 * A379504 A083928 A074038

Adjacent sequences: A379502 A379503 A379504 * A379506 A379507 A379508

Keyword

nonn

Author

Antti Karttunen, Jan 07 2025

Status

approved