A048055 Numbers k such that (sum of the nonprime proper divisors of k) - (sum of prime divisors of k) = k.

532, 945, 2624, 5704, 6536, 229648, 497696, 652970, 685088, 997408, 1481504, 11177984, 32869504, 52813084, 132612224, 224841856, 2140668416, 2404135424, 2550700288, 6469054976, 9367192064, 19266023936, 23414463358, 31381324288, 45812547584, 55620289024

Offset: 1

Author

Naohiro Nomoto

Keywords

• nonn
• nice

Comments

From Peter Luschny, Dec 14 2009: (Start)

A term of this sequence is a Zumkeller number (A083207) since the set of its divisors can be partitioned into two disjoint parts so that the sums of the two parts are equal.

1 + sigma*(k) = sigma'(k) + k

sigma*(k) := Sum_{1 < d < k, d|k, d not prime}, (A060278),

sigma'(k) := Sum_{1 < d < k, d|k, d prime}, (A105221). (End)

Examples

			532 = 1 - 2 + 4 - 7 + 14 - 19 + 28 + 38 + 76 + 133 + 266.
		

Links

• Donovan Johnson, Table of n, a(n) for n = 1..34 (terms <= 10^12)
• Donovan Johnson, 82 terms > 10^12.
• Peter Luschny, Zumkeller Numbers.

Crossrefs

Cf. A083207, A105221, A060278, A000203, A027751, A010051, A002808.

Programs

Haskell

import Data.List (partition)
a048055 n = a048055_list !! (n-1)
a048055_list = [x | x <- a002808_list,
               let (us,vs) = partition ((== 1) . a010051) $ a027751_row x,
               sum us + x == sum vs]
-- Reinhard Zumkeller, Apr 05 2013

Maple

with(numtheory): A048055 := proc(n) local k;
if sigma(n)=2*(n+add(k,k=select(isprime,divisors(n))))
then n else NULL fi end: seq(A048055(i),i=1..7000);
# Peter Luschny, Dec 14 2009

Mathematica

zummableQ[n_] := DivisorSigma[1, n] == 2*(n + Total[Select[Divisors[n], PrimeQ]]); n = 2; A048055 = {}; While[n < 10^6, If[zummableQ[n], Print[n]; AppendTo[A048055, n]]; n++]; A048055 (* Jean-François Alcover, Dec 07 2011, after Peter Luschny *)

Python

from sympy import divisors, primefactors
A048055 = []
for n in range(1,10**4):
    s = sum(divisors(n))
    if not s % 2 and 2*n <= s and (s-2*n)/2 == sum(primefactors(n)):
        A048055.append(n) # Chai Wah Wu, Aug 20 2014

Extensions

a(15)-a(19) from Donovan Johnson, Dec 07 2008

a(20)-a(24) from Donovan Johnson, Jul 06 2010

a(25)-a(26) from Donovan Johnson, Feb 09 2012