A236108 Nonprimes whose proper divisors are partition numbers.

4, 6, 9, 10, 14, 15, 21, 22, 25, 33, 35, 49, 55, 77, 121, 202, 303, 505, 707, 1111, 10201, 35954, 53931, 89885, 125839, 197747, 1815677, 21239726, 31859589, 53099315, 74339041, 116818493, 323172529, 1072606163, 13241661778, 19862492667, 33104154445, 46345816223, 72829139779

Offset: 1

Author

Omar E. Pol, Jan 22 2014

Keywords

• nonn

Comments

Known terms are squares of A049575 or products of 2 distinct terms of A049575. - Michel Marcus, Jan 25 2023

This conjecture holds for terms <= 10^16. - David A. Corneth, Jan 25 2023

Examples

			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.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..81

Crossrefs

Cf. A000041, A018252, A049575, A167392.

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

Programs

Maple

isA000041 := proc(n)
    local k,P;
    for k from 1 do
        P := combinat[numbpart](k) ;
        if P > n then
            return false;
        elif P = n then
            return true ;
        end if;
    end do:
end proc:
isA236108 := proc(n)
    local pdvs,d ;
    if n =1 or isprime(n) then
        return false;
    end if;
    pdvs := numtheory[divisors](n) minus {n} ;
    for d in pdvs do
        if not isA000041(d) then
            return false;
        end if;
    end do:
    return true;
end proc:
for n from 1 to 300000 do
    if isA236108(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jan 29 2014

Mathematica

partitionNumbers = Table[PartitionsP[n], {n, 1, 1000}];
Select[Range[2, 10000],
 If[! PrimeQ[#],
ContainsOnly[Divisors[#][[2 ;; -2]], partitionNumbers]] &] (* Julien Kluge, Dec 03 2016 *)

Extensions

a(17)-a(26) from R. J. Mathar, Jan 29 2014

a(27)-a(32) from Jon E. Schoenfield, Feb 05 2014

a(33)-a(34) from Michel Marcus, Jan 24 2023

More terms from David A. Corneth, Jan 25 2023