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 A345704 Zumkeller numbers k (A083207) such that the next Zumkeller number is k + 12. 1
 282, 840, 1596, 1794, 1920, 2496, 2928, 3108, 3522, 3540, 3594, 4008, 4188, 4602, 4620, 4998, 5268, 5862, 6060, 6708, 6888, 7086, 7788, 7968, 8382, 8400, 9048, 9840, 10362, 10542, 10920, 11100, 11568, 12126, 12162, 12180, 13422, 14106, 14322, 14394, 14880, 15348 (list; graph; refs; listen; history; text; internal format)
 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 A345704_list = list(islice(A345704_gen(), 10)) # Chai Wah Wu, Feb 13 2023 CROSSREFS Cf. A083207, A179529, A328327. Sequence in context: A114804 A250580 A252364 * A224115 A248459 A172627 Adjacent sequences: A345701 A345702 A345703 * A345705 A345706 A345707 KEYWORD nonn AUTHOR Amiram Eldar, Jun 24 2021 STATUS approved

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Last modified August 10 18:57 EDT 2024. Contains 375058 sequences. (Running on oeis4.)