cp's OEIS Frontend

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

%I A048055 #42 May 22 2025 10:21:34
%S A048055 532,945,2624,5704,6536,229648,497696,652970,685088,997408,1481504,
%T A048055 11177984,32869504,52813084,132612224,224841856,2140668416,2404135424,
%U A048055 2550700288,6469054976,9367192064,19266023936,23414463358,31381324288,45812547584,55620289024
%N A048055 Numbers k such that (sum of the nonprime proper divisors of k) - (sum of prime divisors of k) = k.
%C A048055 From _Peter Luschny_, Dec 14 2009: (Start)
%C A048055 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.
%C A048055 1 + sigma*(k) = sigma'(k) + k
%C A048055 sigma*(k) := Sum_{1 < d < k, d|k, d not prime}, (A060278),
%C A048055 sigma'(k) := Sum_{1 < d < k, d|k, d prime}, (A105221). (End)
%H A048055 Donovan Johnson, <a href="/A048055/b048055.txt">Table of n, a(n) for n = 1..34</a> (terms <= 10^12)
%H A048055 Donovan Johnson, <a href="/A048055/a048055.txt">82 terms > 10^12</a>.
%H A048055 Peter Luschny, <a href="http://www.luschny.de/math/seq/ZumkellerNumbers.html"> Zumkeller Numbers</a>.
%e A048055 532 = 1 - 2 + 4 - 7 + 14 - 19 + 28 + 38 + 76 + 133 + 266.
%p A048055 with(numtheory): A048055 := proc(n) local k;
%p A048055 if sigma(n)=2*(n+add(k,k=select(isprime,divisors(n))))
%p A048055 then n else NULL fi end: seq(A048055(i),i=1..7000);
%p A048055 # _Peter Luschny_, Dec 14 2009
%t A048055 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_ *)
%o A048055 (Haskell)
%o A048055 import Data.List (partition)
%o A048055 a048055 n = a048055_list !! (n-1)
%o A048055 a048055_list = [x | x <- a002808_list,
%o A048055                let (us,vs) = partition ((== 1) . a010051) $ a027751_row x,
%o A048055                sum us + x == sum vs]
%o A048055 -- _Reinhard Zumkeller_, Apr 05 2013
%o A048055 (Python)
%o A048055 from sympy import divisors, primefactors
%o A048055 A048055 = []
%o A048055 for n in range(1,10**4):
%o A048055     s = sum(divisors(n))
%o A048055     if not s % 2 and 2*n <= s and (s-2*n)/2 == sum(primefactors(n)):
%o A048055         A048055.append(n) # _Chai Wah Wu_, Aug 20 2014
%Y A048055 Cf. A083207, A105221, A060278, A000203, A027751, A010051, A002808.
%K A048055 nonn,nice
%O A048055 1,1
%A A048055 _Naohiro Nomoto_
%E A048055 a(15)-a(19) from _Donovan Johnson_, Dec 07 2008
%E A048055 a(20)-a(24) from _Donovan Johnson_, Jul 06 2010
%E A048055 a(25)-a(26) from _Donovan Johnson_, Feb 09 2012