A175583 Zumkeller numbers n such that sigma(n)/2 - n is prime.

12, 40, 70, 80, 88, 160, 272, 320, 490, 544, 928, 1184, 1312, 1332, 1575, 1696, 1888, 2420, 2560, 2624, 2628, 3380, 3392, 3712, 3920, 4030, 4100, 4736, 5120, 5248, 5696, 5830, 6464, 6664, 6784, 7232, 7424, 7808, 8228, 8704, 8784, 8925, 9680, 10100

Offset

1,1

Comments

Zumkeller numbers such that A175582(n)-A083207(n)=prime.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Example

a(7)=272 because A175582(57)-A083207(57)=279-272=7 is prime.

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]]; s = Select[ Range@ 10399, ZumkellerQ]; Select[s, PrimeQ[ DivisorSigma[1, # ]/2 - # ] &] (* Robert G. Wilson v, Aug 03 2010 *)

Prog

(Python)
from sympy import isprime, divisors
from sympy.combinatorics.subsets import Subset
for n in range(1, 10**5):
    d = divisors(n)
    s = sum(d)
    if not s % 2 and max(d)<= s//2 and isprime(s//2-n):
        for x in range(1, 2**len(d)):
            if sum(Subset.unrank_binary(x, d).subset) == s//2:
                print(n, end=', ')
                break
# Chai Wah Wu, Aug 13 2014
(Python)
from sympy import isprime, divisors
import numpy as np
A175583 = []
for n in range(1, 10**5):
    d = divisors(n)
    s = sum(d)
    if not s % 2 and 2*n <= s and isprime(s//2-n):
        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:
                A175583.append(n)
                break
# Chai Wah Wu, Aug 20 2014

Crossrefs

Cf. A083207.

Sequence in context: A114815 A363121 A353839 * A109766 A365446 A033586

Adjacent sequences: A175580 A175581 A175582 * A175584 A175585 A175586

Keyword

nonn

Author

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

Extensions

a(17) - a(44) from Robert G. Wilson v, Aug 03 2010

Definition and example corrected by Chai Wah Wu, Aug 13 2014

Status

approved