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

1,1

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.

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 *)

Prog

(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

Crossrefs

Cf. A083207, A023196.

Sequence in context: A295984 A218596 A043633 * A251814 A320714 A251647

Adjacent sequences: A171638 A171639 A171640 * A171642 A171643 A171644

Keyword

nonn

Author

Peter Luschny, Dec 14 2009

Status

approved