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%I #26 Nov 19 2019 06:54:14
%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**5):
%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
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