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A175592 Numbers n whose prime factors can be partitioned into two disjoint sets whose sums are both (sum of primes dividing n (with repetition))/2.

%I A175592 #30 Oct 21 2024 13:12:32
%S A175592 4,9,16,25,30,36,49,64,70,72,81,84,100,120,121,144,169,196,225,240,
%T A175592 256,270,280,286,288,289,308,324,336,361,378,400,440,441,480,484,495,
%U A175592 525,528,529,540,576,594,625,630,646,648,672,676,728,729,750,756,784,800
%N A175592 Numbers n whose prime factors can be partitioned into two disjoint sets whose sums are both (sum of primes dividing n (with repetition))/2.
%C A175592 From _Christian N. K. Anderson_, Apr 16 2013: (Start)
%C A175592 Alternatively, the two sets of prime factors have an equal sum.
%C A175592 Superset of even powers, p^(2*i) where p is a prime number (A056798), and composites thereof. (End)
%H A175592 Christian N. K. Anderson, <a href="/A175592/b175592.txt">Table of n, a(n) for n = 1..10000</a>
%H A175592 Christian N. K. Anderson, <a href="/A175592/a175592_1.txt">Equal sum partitions</a> of prime factors of a(n).
%e A175592 a(1)=4 because 4=2*2 and 2=2, a(2)=9 because 9=3*3 and 3=3, a(3)=16 because 16=2*2*2*2 and 2+2=2+2, a(4)=25 because 25=5*5 and 5=5, a(5)=30 because 30=2*3*5 and 2+3=5.
%p A175592 Primefactors := proc(n) local F, f, i; F := [];
%p A175592 for f in ifactors(n)[2] do
%p A175592     for i from 1 to f[2] do
%p A175592         F := [op(F), f[1]]
%p A175592 od od; F end:
%p A175592 isPrimeZumkeller := proc(n) option remember; local s, p, i, P;
%p A175592     s := add(Primefactors(n));  # A001414
%p A175592     if s::odd or s = 0 then return false fi;
%p A175592     P := mul(1 + x^i, i in Primefactors(n));
%p A175592     is(0 < coeff(P, x, s/2)) end:
%p A175592 select(isPrimeZumkeller, [seq(1..800)]);   # _Peter Luschny_, Oct 21 2024
%o A175592 (Haskell)
%o A175592 a175592 n = a175592_list !! (n-1)
%o A175592 a175592_list = filter (z 0 0 . a027746_row) $ [1..] where
%o A175592    z u v []     = u == v
%o A175592    z u v (p:ps) = z (u + p) v ps || z u (v + p) ps
%o A175592 -- _Reinhard Zumkeller_, Apr 18 2013
%Y A175592 Prime divisor variant of A083207.
%Y A175592 Cf. A001414, A056798, A027746, A221054, A001414.
%K A175592 nonn
%O A175592 1,1
%A A175592 _Juri-Stepan Gerasimov_, Jul 20 2010
%E A175592 Corrected by _Christian N. K. Anderson_, Apr 16 2013