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A379503 Almost Zumkeller numbers: Numbers whose Zumkeller-deficiency (A103977) is 1.

%I A379503 #38 Jan 07 2025 08:47:43
%S A379503 1,2,4,8,16,18,32,36,64,72,100,128,144,162,196,200,256,288,324,392,
%T A379503 400,450,512,576,648,784,800,882,900,968,1024,1152,1296,1352,1458,
%U A379503 1568,1600,1764,1800,1936,2048,2178,2304,2450,2500,2592,2704,2916,3042,3136,3200,3528,3600,3872,4050,4096,4356,4608,4624,4900,5000
%N A379503 Almost Zumkeller numbers: Numbers whose Zumkeller-deficiency (A103977) is 1.
%C A379503 Numbers whose divisors can be partitioned into two disjoint sets with equal sum when an extra 1-divisor is added to them. - _Amiram Eldar_, Jan 06 2025
%C A379503 Question: Does A156942 give all odd squares > 1 of this sequence? There are two issues here: first, whether there are any almost perfect numbers (k such that sigma(k) = 2k-1) that are odd (and by necessity squares) other than 1, and second, whether A103977(k) = 1 for all terms of A156942? The first 15000 terms of A156942 are all members.
%H A379503 Robert Israel, <a href="/A379503/b379503.txt">Table of n, a(n) for n = 1..2500</a> (n = 1..430 from Antti Karttunen)
%e A379503 18 is included, as its divisors with an extra 1 are [1, 1, 2, 3, 6, 9, 18], and these can be partitioned as 2+3+6+9 = 1+1+18 = 20.
%e A379503 36 is included, as its divisors with an extra 1 are [1, 1, 2, 3, 4, 6, 9, 12, 18, 36], and these can be partitioned to two sets with equal sums, for example as (1+2+3+4)+(36) = (1+9)+(6+12+18), and also in several other ways (see example in A379504).
%e A379503 11025 is included as its divisors with an extra 1 are [1, 1, 3, 5, 7, 9, 15, 21, 25, 35, 45, 49, 63, 75, 105, 147, 175, 225, 245, 315, 441, 525, 735, 1225, 1575, 2205, 3675, 11025], and 1+5+35+175+245+11025 = 1+3+7+9+15+21+25+45+49+63+75+105+147+225+315+441+525+735+1225+1575+2205+3675 = 11486 = (sigma(11025)+1)/2.
%p A379503 KK:= proc(S) # Karmarkar-Karp algorithm
%p A379503   local R,n,a,b;
%p A379503   R:= S;
%p A379503   for n from nops(R) by -1 to 2 do
%p A379503     R:= sort([abs(R[-1]-R[-2]), op(R[1..-3])]);
%p A379503   od;
%p A379503   op(R) = 0
%p A379503 end proc:
%p A379503 filter:= proc(n) local S,t,d,R,i;
%p A379503   S:= [1, op(numtheory:-divisors(n))];
%p A379503   t:= convert(S,`+`)/2;
%p A379503   if t < n then return false fi;
%p A379503   if not t::integer then return false fi;
%p A379503   if KK(S) then return true fi;
%p A379503   evalb(coeff(mul(1+x^d,d=S),x,t) <> 0)
%p A379503 end proc;
%p A379503 select(filter, [$1..10000]); # _Robert Israel_, Jan 06 2025
%o A379503 (PARI) is_A379503 = A379502;
%Y A379503 Positions of 1's in A103977, positions of nonzero terms in A379504.
%Y A379503 Cf. A083207, A379502 (characteristic function).
%Y A379503 Subsequences: A000079, A156942 (conjectured)
%Y A379503 Subsequence of A028982, and of A083210.
%K A379503 nonn
%O A379503 1,2
%A A379503 _Antti Karttunen_, Jan 06 2025