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%I #40 Sep 18 2023 02:06:23
%S 532,945,2624,5704,6536,229648,497696,652970,685088,997408,1481504,
%T 11177984,32869504,52813084,132612224,224841856,2140668416,2404135424,
%U 2550700288,6469054976,9367192064,19266023936,23414463358,31381324288,45812547584,55620289024
%N Numbers k such that (sum of the nonprime proper divisors of k) - (sum of prime divisors of k) = k.
%C From _Peter Luschny_, Dec 14 2009: (Start)
%C 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 1 + sigma*(k) = sigma'(k) + k
%C sigma*(k) := Sum_{1 < d < k, d|k, d not prime}, (A060278),
%C sigma'(k) := Sum_{1 < d < k, d|k, d prime}, (A105221). (End)
%H Donovan Johnson, <a href="/A048055/b048055.txt">Table of n, a(n) for n = 1..34</a> (terms <= 10^12)
%H Donovan Johnson, <a href="/A048055/a048055.txt">82 terms > 10^12</a>.
%H Peter Luschny, <a href="http://www.luschny.de/math/seq/ZumkellerNumbers.html"> Zumkeller Numbers</a>.
%e 532 = 1 - 2 + 4 - 7 + 14 - 19 + 28 + 38 + 76 + 133 + 266.
%p with(numtheory): A048055 := proc(n) local k;
%p if sigma(n)=2*(n+add(k,k=select(isprime,divisors(n))))
%p then n else NULL fi end: seq(A048055(i),i=1..7000);
%p # _Peter Luschny_, Dec 14 2009
%t 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 (Haskell)
%o import Data.List (partition)
%o a048055 n = a048055_list !! (n-1)
%o a048055_list = [x | x <- a002808_list,
%o let (us,vs) = partition ((== 1) . a010051) $ a027751_row x,
%o sum us + x == sum vs]
%o -- _Reinhard Zumkeller_, Apr 05 2013
%o (Python)
%o from sympy import divisors, primefactors
%o A048055 = []
%o for n in range(1,10**4):
%o ....s = sum(divisors(n))
%o ....if not s % 2 and 2*n <= s and (s-2*n)/2 == sum(primefactors(n)):
%o ........A048055.append(n) # _Chai Wah Wu_, Aug 20 2014
%Y Cf. A083207, A105221, A060278, A000203, A027751, A010051, A002808.
%K nonn,nice
%O 1,1
%A _Naohiro Nomoto_
%E a(15)-a(19) from _Donovan Johnson_, Dec 07 2008
%E a(20)-a(24) from _Donovan Johnson_, Jul 06 2010
%E a(25)-a(26) from _Donovan Johnson_, Feb 09 2012
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