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A360694 Numbers whose divisors can be partitioned into two disjoint sets where the sum of both sets is prime.

%I A360694 #74 Mar 21 2023 18:07:47
%S A360694 4,6,8,10,12,14,15,16,18,20,21,22,24,26,27,28,30,32,33,34,35,36,38,39,
%T A360694 40,42,44,45,48,52,54,55,56,57,58,60,63,65,66,68,69,70,72,75,76,77,78,
%U A360694 80,82,84,85,86,88,90,92,93,94,95,96,99,102,104,105,106,108,110,111,112,114,115
%N A360694 Numbers whose divisors can be partitioned into two disjoint sets where the sum of both sets is prime.
%C A360694 The concept of this sequence is similar to the concept of Zumkeller numbers (A083207) partitioning the sums of the divisors (A000203) into two sets.
%C A360694 This concept can be extended, since the sums of some numbers' divisors can be partitioned into more sets, e.g., 6 (2,3,7) and 10 (2,5,11) into three.
%C A360694 Some numbers can be divided more than one way. For 10, there are two divisons: (5,13) and (7,11) and for 20, there are four: (5,37), (11,31), (13,29) and (19,23).
%C A360694 From _Robert Israel_, Feb 21 2023: (Start)
%C A360694 Contains no primes.
%C A360694 k in A028982 is in the sequence iff k is even and A000203(k)-2 is prime.
%C A360694 (End)
%D A360694 Song Y. Yan, Perfect, Amicable and Sociable Numbers, World Scientific Pub Co Inc., 1996, p. 11, p. 22.
%H A360694 Yuejian Peng and K. P. S. Bhaskara Rao, <a href="https://doi.org/10.1016/j.jnt.2012.09.020">On Zumkeller numbers</a>, Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155
%H A360694 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PerfectNumber.html">Perfect Number</a>.
%H A360694 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Prime.html">Prime</a>.
%e A360694 4 belongs to the sequence, since its divisors can be partitioned into two sets where the sums of these sets are primes (2,5). 9 does not belong to the sequence, because its divisors cannot be partitioned in this way.
%p A360694 filter:= proc(n) local P,p,S,s;
%p A360694   S:= numtheory:-divisors(n); s:= convert(S,`+`);
%p A360694   P:= combinat:-subsets(S minus {n});
%p A360694   while not P[finished] do
%p A360694     p:= convert(P[nextvalue](),`+`);
%p A360694     if isprime(p) and isprime(s-p) then return true fi
%p A360694   od;
%p A360694   false
%p A360694 end proc:
%p A360694 select(filter, [$1..200]); # _Robert Israel_, Feb 21 2023
%t A360694 q[n_] := Module[{d = Divisors[n], s, p}, s = Total[d]; p = Position[Rest @ CoefficientList[Product[1 + x^i, {i, d}], x], _?(# > 0 &)] // Flatten; AnyTrue[p, PrimeQ[#] && PrimeQ[s - #] &]]; Select[Range[100], q] (* _Amiram Eldar_, Feb 18 2023 *)
%Y A360694 Cf. A000040, A000396, A028982, A083207, A000203.
%K A360694 nonn
%O A360694 1,1
%A A360694 _Zoltan Galantai_, Feb 17 2023