A175582 a(n) = sigma(n-th Zumkeller number)/2.

6, 14, 21, 30, 28, 36, 45, 48, 62, 60, 60, 84, 72, 72, 84, 93, 112, 90, 117, 126, 108, 105, 140, 124, 120, 180, 156, 168, 144, 168, 186, 196, 189, 240, 180, 186, 273, 192, 254, 234, 252, 217, 288, 300, 252, 228, 252, 280, 273, 372, 252, 364, 264, 294, 360, 360, 279

Offset

1,1

Comments

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

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Formula

a(n) = A000203(A083207(n))/2. - Michel Marcus, Aug 21 2014

Mathematica

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 *)

Prog

(Python)
from sympy import divisors
import numpy as np
A175582 = []
for n in range(1, 10**3):
    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[i, s2] == s2:
                A175582.append(int(s/2))
                break
# Chai Wah Wu, Aug 21 2014

Crossrefs

Cf. A083207, A000203.

Sequence in context: A184924 A110223 A190504 * A182081 A125086 A195063

Adjacent sequences: A175579 A175580 A175581 * A175583 A175584 A175585

Keyword

nonn

Author

Vladislav-Stepan Malakhovsky and Juri-Stepan Gerasimov, Jul 15 2010

Extensions

Inserted a(45) and corrected typo in a(49) and crossrefs by Chai Wah Wu, Aug 21 2014

Status

approved