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A175582 a(n) = sigma(n-th Zumkeller number)/2. 3
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 (list; graph; refs; listen; history; text; internal format)
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**5):

....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

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Last modified October 2 22:14 EDT 2022. Contains 357230 sequences. (Running on oeis4.)