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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.

%I A379505 #21 Jun 02 2025 12:12:29
%S A379505 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 A379505 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,
%U A379505 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
%N 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.
%C A379505 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).
%C A379505 Conjecture 2: a(n) = 1 if and only if n = 2^k, with k >= 0. This claim is equal 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.
%H A379505 Antti Karttunen, <a href="/A379505/b379505.txt">Table of n, a(n) for n = 1..1799</a> [a larger b-file requested]
%F A379505 a(n) <= A379504(n).
%F A379505 A103977(n) = 1 <=> a(n) > 0.
%e A379505 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.
%e A379505 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:
%e A379505   1+1+2+6+36  = 3+4+9+12+18,
%e A379505   1+2+3+4+36  = 1+6+9+12+18,
%e A379505   1+3+6+36    = 1+2+4+9+12+18,
%e A379505   1+9+36      = 1+2+3+4+6+12+18,
%e A379505   4+6+36      = 1+1+2+3+9+12+18,
%e A379505 where each sum on the left and right hand side gives (sigma(36)+1)/2 = 46.
%e A379505 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:
%e A379505   [2, 24, 72],
%e A379505   [1, 1, 24, 72],
%e A379505   [8, 18, 72],
%e A379505   [2, 6, 18, 72],
%e A379505   [1, 1, 6, 18, 72],
%e A379505   [1, 3, 4, 18, 72],
%e A379505   [1, 1, 2, 4, 18, 72],
%e A379505   [1, 4, 9, 12, 72],
%e A379505   [2, 3, 9, 12, 72],
%e A379505   [1, 1, 3, 9, 12, 72],
%e A379505   [6, 8, 12, 72],
%e A379505   [2, 4, 8, 12, 72],
%e A379505   [1, 1, 4, 8, 12, 72],
%e A379505   [1, 2, 3, 8, 12, 72],
%e A379505   [1, 3, 4, 6, 12, 72],
%e A379505   [1, 1, 2, 4, 6, 12, 72],
%e A379505   [3, 6, 8, 9, 72],
%e A379505   [1, 2, 6, 8, 9, 72],
%e A379505   [2, 3, 4, 8, 9, 72],
%e A379505   [1, 1, 3, 4, 8, 9, 72],
%e A379505   [1, 1, 2, 3, 4, 6, 9, 72],
%e A379505   [8, 12, 18, 24, 36],
%e A379505   [2, 6, 12, 18, 24, 36],
%e A379505   [1, 1, 6, 12, 18, 24, 36],
%e A379505   [1, 3, 4, 12, 18, 24, 36],
%e A379505   [1, 1, 2, 4, 12, 18, 24, 36],
%e A379505   [3, 8, 9, 18, 24, 36],
%e A379505   [1, 2, 8, 9, 18, 24, 36],
%e A379505   [1, 4, 6, 9, 18, 24, 36],
%e A379505   [2, 3, 6, 9, 18, 24, 36],
%e A379505   [1, 1, 3, 6, 9, 18, 24, 36],
%e A379505   [1, 1, 2, 3, 4, 9, 18, 24, 36],
%e A379505   [2, 4, 6, 8, 18, 24, 36],
%e A379505   [1, 1, 4, 6, 8, 18, 24, 36],
%e A379505   [1, 2, 3, 6, 8, 18, 24, 36],
%e A379505   [3, 6, 8, 9, 12, 24, 36],
%e A379505   [1, 2, 6, 8, 9, 12, 24, 36],
%e A379505   [2, 3, 4, 8, 9, 12, 24, 36],
%e A379505   [1, 1, 3, 4, 8, 9, 12, 24, 36],
%e A379505   [1, 1, 2, 3, 4, 6, 9, 12, 24, 36],
%e A379505   [2, 3, 4, 6, 8, 9, 12, 18, 36],
%e A379505   [1, 1, 3, 4, 6, 8, 9, 12, 18, 36],
%e A379505 therefore a(72) = 42/2 = 21.
%o A379505 (PARI)
%o A379505 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)));
%o A379505 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));
%Y A379505 Cf. A083206, A103977, A156942, A336700, A379502, A379503 (positions of nonzero terms), A379504 (version where two 1's are considered distinct).
%Y A379505 Cf. A000079 (conjectured to give the positions of 1's).
%K A379505 nonn
%O A379505 1,18
%A A379505 _Antti Karttunen_, Jan 07 2025