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

1,2

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.

Links

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

Example

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.

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

Prog

(PARI) is_A379503 = A379502;

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.

Sequence in context: A072462 A369951 A088827 * A316900 A076057 A364061

Adjacent sequences: A379500 A379501 A379502 * A379504 A379505 A379506

Keyword

nonn

Author

Antti Karttunen, Jan 06 2025

Status

approved