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A236108 Nonprimes whose proper divisors are partition numbers.

%I A236108 #53 Jan 25 2023 15:59:38
%S A236108 4,6,9,10,14,15,21,22,25,33,35,49,55,77,121,202,303,505,707,1111,
%T A236108 10201,35954,53931,89885,125839,197747,1815677,21239726,31859589,
%U A236108 53099315,74339041,116818493,323172529,1072606163,13241661778,19862492667,33104154445,46345816223,72829139779
%N A236108 Nonprimes whose proper divisors are partition numbers.
%C A236108 Known terms are squares of A049575 or products of 2 distinct terms of A049575. - _Michel Marcus_, Jan 25 2023
%C A236108 This conjecture holds for terms <= 10^16. - _David A. Corneth_, Jan 25 2023
%H A236108 David A. Corneth, <a href="/A236108/b236108.txt">Table of n, a(n) for n = 1..81</a>
%e A236108 10 is in the sequence because 10 is a nonprime number and the proper divisors of 10 are 1, 2, 5, which are also partition numbers.
%p A236108 isA000041 := proc(n)
%p A236108     local k,P;
%p A236108     for k from 1 do
%p A236108         P := combinat[numbpart](k) ;
%p A236108         if P > n then
%p A236108             return false;
%p A236108         elif P = n then
%p A236108             return true ;
%p A236108         end if;
%p A236108     end do:
%p A236108 end proc:
%p A236108 isA236108 := proc(n)
%p A236108     local pdvs,d ;
%p A236108     if n =1 or isprime(n) then
%p A236108         return false;
%p A236108     end if;
%p A236108     pdvs := numtheory[divisors](n) minus {n} ;
%p A236108     for d in pdvs do
%p A236108         if not isA000041(d) then
%p A236108             return false;
%p A236108         end if;
%p A236108     end do:
%p A236108     return true;
%p A236108 end proc:
%p A236108 for n from 1 to 300000 do
%p A236108     if isA236108(n) then
%p A236108         printf("%d,",n) ;
%p A236108     end if;
%p A236108 end do: # _R. J. Mathar_, Jan 29 2014
%t A236108 partitionNumbers = Table[PartitionsP[n], {n, 1, 1000}];
%t A236108 Select[Range[2, 10000],
%t A236108  If[! PrimeQ[#],
%t A236108 ContainsOnly[Divisors[#][[2 ;; -2]], partitionNumbers]] &] (* _Julien Kluge_, Dec 03 2016 *)
%Y A236108 Cf. A000041, A018252, A049575, A167392.
%Y A236108 Cf. A236102, A236103, A236105, A236107, A236110.
%K A236108 nonn
%O A236108 1,1
%A A236108 _Omar E. Pol_, Jan 22 2014
%E A236108 a(17)-a(26) from _R. J. Mathar_, Jan 29 2014
%E A236108 a(27)-a(32) from _Jon E. Schoenfield_, Feb 05 2014
%E A236108 a(33)-a(34) from _Michel Marcus_, Jan 24 2023
%E A236108 More terms from _David A. Corneth_, Jan 25 2023