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

4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 52, 54, 55, 56, 57, 58, 60, 63, 65, 66, 68, 69, 70, 72, 75, 76, 77, 78, 80, 82, 84, 85, 86, 88, 90, 92, 93, 94, 95, 96, 99, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115

Offset: 1

Author

Zoltan Galantai, Feb 17 2023

Keywords

• nonn

Comments

The concept of this sequence is similar to the concept of Zumkeller numbers (A083207) partitioning the sums of the divisors (A000203) into two sets.

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.

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).

From Robert Israel, Feb 21 2023: (Start)

Contains no primes.

k in A028982 is in the sequence iff k is even and A000203(k)-2 is prime.

(End)

Examples

			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.
		

References

• Song Y. Yan, Perfect, Amicable and Sociable Numbers, World Scientific Pub Co Inc., 1996, p. 11, p. 22.

Links

• Yuejian Peng and K. P. S. Bhaskara Rao, On Zumkeller numbers, Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155
• Eric Weisstein's World of Mathematics, Perfect Number.
• Eric Weisstein's World of Mathematics, Prime.

Crossrefs

Cf. A000040, A000396, A028982, A083207, A000203.

Programs

Maple

filter:= proc(n) local P,p,S,s;
  S:= numtheory:-divisors(n); s:= convert(S,`+`);
  P:= combinat:-subsets(S minus {n});
  while not P[finished] do
    p:= convert(P[nextvalue](),`+`);
    if isprime(p) and isprime(s-p) then return true fi
  od;
  false
end proc:
select(filter, [$1..200]); # Robert Israel, Feb 21 2023

Mathematica

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 *)