A171641 Non-deficient numbers with even sigma which are not Zumkeller.

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, 4014, 4068, 4086

Offset: 1

Author

Peter Luschny, Dec 14 2009

Keywords

• nonn

Comments

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.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Chai Wah Wu)
• Peter Luschny, Zumkeller Numbers.

Crossrefs

Cf. A000203, A023196, A083207.

Setwise difference A083211 \ A156903.

Positions of even negative terms in A378600.

Programs

Maple

with(NumberTheory):
isA171641 := proc(n) local s, p, i, P;
    s := SumOfDivisors(n);
    if s::odd or s < n*2 then false else
    P := mul(1 + x^i, i in Divisors(n));
    0 = coeff(P, x, s/2) fi end:
select(isA171641, [seq(1..4100)]);  # Peter Luschny, Oct 19 2024

Mathematica

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 *)

Python

from sympy import divisors
import numpy as np
A171641 = []
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:
            A171641.append(n)
# Chai Wah Wu, Aug 19 2014