A175582 - OEIS

%I #29 Oct 18 2024 18:00:22

%S 6,14,21,30,28,36,45,48,62,60,60,84,72,72,84,93,112,90,117,126,108,

%T 105,140,124,120,180,156,168,144,168,186,196,189,240,180,186,273,192,

%U 254,234,252,217,288,300,252,228,252,280,273,372,252,364,264,294,360,360,279

%N a(n) = sigma(n-th Zumkeller number)/2.

%C Conjecture: Any 4 consecutive terms include at least one Zumkeller number (verified for the first 10^5 terms). - _Ivan N. Ianakiev_, Oct 17 2019

%H Chai Wah Wu, <a href="/A175582/b175582.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A000203(A083207(n))/2. - _Michel Marcus_, Aug 21 2014

%t ZumkellerQ[n_] := Module[{d = Divisors[n], t, ds, x}, ds = Plus @@ d; If[ Mod[ds, 2] > 0, False, t = CoefficientList[ Product[1 + x^i, {i, d}], x]; t[[1 + ds/2]] > 0]]; DivisorSigma[1, Select[ Range@ 275, ZumkellerQ]]/2 (* _Robert G. Wilson v_, Aug 03 2010 *)

%o (Python)

%o from sympy import divisors

%o import numpy as np

%o A175582 = []

%o for n in range(1, 10**3):

%o     d = divisors(n)

%o     s = sum(d)

%o     if not s % 2 and 2*n <= s:

%o         d.remove(n)

%o         s2, ld = int(s/2-n), len(d)

%o         z = np.zeros((ld+1, s2+1), dtype=int)

%o         for i in range(1, ld+1):

%o             y = min(d[i-1], s2+1)

%o             z[i, range(y)] = z[i-1, range(y)]

%o             z[i, range(y, s2+1)] = np.maximum(z[i-1, range(y, s2+1)], z[i-1, range(0, s2+1-y)]+y)

%o             if z[i, s2] == s2:

%o                 A175582.append(int(s/2))

%o                 break

%o # _Chai Wah Wu_, Aug 21 2014

%Y Cf. A083207, A000203.

%K nonn

%O 1,1

%A Vladislav-Stepan Malakhovsky and _Juri-Stepan Gerasimov_, Jul 15 2010

%E Inserted a(45) and corrected typo in a(49) and crossrefs by _Chai Wah Wu_, Aug 21 2014