A236108 - OEIS

%I #53 Jan 25 2023 15:59:38

%S 4,6,9,10,14,15,21,22,25,33,35,49,55,77,121,202,303,505,707,1111,

%T 10201,35954,53931,89885,125839,197747,1815677,21239726,31859589,

%U 53099315,74339041,116818493,323172529,1072606163,13241661778,19862492667,33104154445,46345816223,72829139779

%N Nonprimes whose proper divisors are partition numbers.

%C Known terms are squares of A049575 or products of 2 distinct terms of A049575. - _Michel Marcus_, Jan 25 2023

%C This conjecture holds for terms <= 10^16. - _David A. Corneth_, Jan 25 2023

%H David A. Corneth, <a href="/A236108/b236108.txt">Table of n, a(n) for n = 1..81</a>

%e 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 isA000041 := proc(n)

%p     local k,P;

%p     for k from 1 do

%p         P := combinat[numbpart](k) ;

%p         if P > n then

%p             return false;

%p         elif P = n then

%p             return true ;

%p         end if;

%p     end do:

%p end proc:

%p isA236108 := proc(n)

%p     local pdvs,d ;

%p     if n =1 or isprime(n) then

%p         return false;

%p     end if;

%p     pdvs := numtheory[divisors](n) minus {n} ;

%p     for d in pdvs do

%p         if not isA000041(d) then

%p             return false;

%p         end if;

%p     end do:

%p     return true;

%p end proc:

%p for n from 1 to 300000 do

%p     if isA236108(n) then

%p         printf("%d,",n) ;

%p     end if;

%p end do: # _R. J. Mathar_, Jan 29 2014

%t partitionNumbers = Table[PartitionsP[n], {n, 1, 1000}];

%t Select[Range[2, 10000],

%t  If[! PrimeQ[#],

%t ContainsOnly[Divisors[#][[2 ;; -2]], partitionNumbers]] &] (* _Julien Kluge_, Dec 03 2016 *)

%Y Cf. A000041, A018252, A049575, A167392.

%Y Cf. A236102, A236103, A236105, A236107, A236110.

%K nonn

%O 1,1

%A _Omar E. Pol_, Jan 22 2014

%E a(17)-a(26) from _R. J. Mathar_, Jan 29 2014

%E a(27)-a(32) from _Jon E. Schoenfield_, Feb 05 2014

%E a(33)-a(34) from _Michel Marcus_, Jan 24 2023

%E More terms from _David A. Corneth_, Jan 25 2023