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A171641
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Non-deficient numbers with even sigma which are not Zumkeller.
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12
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738, 748, 774, 846, 954, 1062, 1098, 1206, 1278, 1314, 1422, 1494, 1602, 1746, 1818, 1854, 1926, 1962, 2034, 2286, 2358, 2466, 2502, 2682, 2718, 2826, 2934, 3006, 3114, 3222, 3258, 3438, 3474, 3492, 3546, 3582, 3636, 3708, 3798, 3852, 3924
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OFFSET
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1,1
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COMMENTS
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Numbers which are non-deficient (sigma(n) >= 2n) [A023196] such that sigma(n) [A000203] is even but which are not Zumkeller numbers [A083207], i.e., the positive factors of n cannot be partitioned into two disjoint parts so that the sums of the two parts are equal.
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LINKS
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MATHEMATICA
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Reap[For[n = 2, n <= 4000, n = n+2, sigma = DivisorSigma[1, n]; If[sigma >= 2n && EvenQ[sigma] && Coefficient[ Times @@ (1 + x^Divisors[n]) // Expand, x, sigma/2] == 0, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jul 26 2013 *)
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PROG
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(Python)
from sympy import divisors
import numpy as np
for n in range(2, 10**6):
....d = divisors(n)
....s = sum(d)
....if not s % 2 and 2*n <= s:
........d.remove(n)
........s2, ld = int(s/2-n), len(d)
........z = np.zeros((ld+1, s2+1), dtype=int)
........for i in range(1, ld+1):
............y = min(d[i-1], s2+1)
............z[i, range(y)] = z[i-1, range(y)]
............z[i, range(y, s2+1)] = np.maximum(z[i-1, range(y, s2+1)], z[i-1, range(0, s2+1-y)]+y)
........if z[ld, s2] != s2:
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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