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.

4, 9, 16, 25, 30, 36, 49, 64, 70, 72, 81, 84, 100, 120, 121, 144, 169, 196, 225, 240, 256, 270, 280, 286, 288, 289, 308, 324, 336, 361, 378, 400, 440, 441, 480, 484, 495, 525, 528, 529, 540, 576, 594, 625, 630, 646, 648, 672, 676, 728, 729, 750, 756, 784, 800

Offset: 1

Author

Juri-Stepan Gerasimov, Jul 20 2010

Keywords

• nonn

Comments

From Christian N. K. Anderson, Apr 16 2013: (Start)

Alternatively, the two sets of prime factors have an equal sum.

Superset of even powers, p^(2*i) where p is a prime number (A056798), and composites thereof. (End)

Examples

			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.
		

Links

• Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
• Christian N. K. Anderson, Equal sum partitions of prime factors of a(n).

Crossrefs

Prime divisor variant of A083207.

Cf. A001414, A056798, A027746, A221054, A001414.

Programs

Haskell

a175592 n = a175592_list !! (n-1)
a175592_list = filter (z 0 0 . a027746_row) $ [1..] where
   z u v []     = u == v
   z u v (p:ps) = z (u + p) v ps || z u (v + p) ps
-- Reinhard Zumkeller, Apr 18 2013

Maple

Primefactors := proc(n) local F, f, i; F := [];
for f in ifactors(n)[2] do
    for i from 1 to f[2] do
        F := [op(F), f[1]]
od od; F end:
isPrimeZumkeller := proc(n) option remember; local s, p, i, P;
    s := add(Primefactors(n));  # A001414
    if s::odd or s = 0 then return false fi;
    P := mul(1 + x^i, i in Primefactors(n));
    is(0 < coeff(P, x, s/2)) end:
select(isPrimeZumkeller, [seq(1..800)]);   # Peter Luschny, Oct 21 2024

Extensions

Corrected by Christian N. K. Anderson, Apr 16 2013