OFFSET
1,1
COMMENTS
Frank Buss and T. D. Noe conjectured (see A083207) and Robert Gerbicz proved that the largest possible gap between Zumkeller numbers is 12 (SeqFan post, 2010). A proof was also published by Mahanta et al. (2020).
LINKS
Robert Israel, Table of n, a(n) for n = 1..648
Robert Gerbicz, A083207 On an observation of Frank Buss, posts to the SeqFan list, July 2010.
Pankaj Jyoti Mahanta, Manjil P. Saikia and Daniel Yaqubi, Some properties of Zumkeller numbers and k-layered numbers, Journal of Number Theory, Vol. 217 (2020), pp. 218-236.
EXAMPLE
282 is a term since it is a Zumkeller number, and the next Zumkeller number is 282 + 12 = 294.
MAPLE
iszum:= proc(n) local D, s, P, d;
D:= numtheory:-divisors(n);
s:= convert(D, `+`);
if s::odd then return false fi;
P:= mul(1+x^d, d=D);
coeff(P, x, s/2) > 0
end proc:
last:= 6: R:= NULL: count:= 0:
for i from 7 while count < 60 do
if iszum(i) then
if i-last = 12 then R:= R, last; count:= count+1 fi;
last:= i;
fi
od:
R; # Robert Israel, Feb 13 2023
MATHEMATICA
zumQ[n_] := Module[{d = Divisors[n], sum, x}, sum = Plus @@ d; EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]; z = Select[Range[5000], zumQ]; z[[Position[Differences[z], 12] // Flatten]]
PROG
(Python)
from itertools import count, islice
from sympy import divisors
def A345704_gen(startvalue=1): # generator of terms >= startvalue
m = -20
for n in count(max(startvalue, 1)):
d = divisors(n)
s = sum(d)
if s&1^1 and n<<1<=s:
d = d[:-1]
s2, ld = (s>>1)-n, len(d)
z = [[0 for _ in range(s2+1)] for _ in range(ld+1)]
for i in range(1, ld+1):
y = min(d[i-1], s2+1)
z[i][:y] = z[i-1][:y]
for j in range(y, s2+1):
z[i][j] = max(z[i-1][j], z[i-1][j-y]+y)
if z[i][s2] == s2:
if m == n-12:
yield m
m = n
break
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jun 24 2021
STATUS
approved