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

1, 2, 4, 8, 16, 18, 32, 36, 64, 72, 100, 128, 144, 162, 196, 200, 256, 288, 324, 392, 400, 450, 512, 576, 648, 784, 800, 882, 900, 968, 1024, 1152, 1296, 1352, 1458, 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

Offset: 1

Author

Antti Karttunen, Jan 06 2025

Keywords

• nonn

Comments

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

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.

Examples

			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.
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).
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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..2500 (n = 1..430 from Antti Karttunen)

Crossrefs

Positions of 1's in A103977, positions of nonzero terms in A379504.

Cf. A083207, A379502 (characteristic function).

Subsequences: A000079, A156942 (conjectured)

Subsequence of A028982, and of A083210.

Programs

Maple

KK:= proc(S) # Karmarkar-Karp algorithm
  local R,n,a,b;
  R:= S;
  for n from nops(R) by -1 to 2 do
    R:= sort([abs(R[-1]-R[-2]), op(R[1..-3])]);
  od;
  op(R) = 0
end proc:
filter:= proc(n) local S,t,d,R,i;
  S:= [1, op(numtheory:-divisors(n))];
  t:= convert(S,`+`)/2;
  if t < n then return false fi;
  if not t::integer then return false fi;
  if KK(S) then return true fi;
  evalb(coeff(mul(1+x^d,d=S),x,t) <> 0)
end proc;
select(filter, [$1..10000]); # Robert Israel, Jan 06 2025

PARI

is_A379503 = A379502;