A195332 Numbers such that the sum of the cube of the odd divisors is prime.

9, 18, 36, 72, 121, 144, 242, 288, 484, 576, 968, 1152, 1936, 2304, 3872, 4608, 7744, 9216, 15488, 18432, 30976, 36481, 36864, 61952, 72361, 72962, 73728, 123904, 144722, 145924, 146689, 147456, 247808, 259081, 289444, 291848, 293378, 294912

Offset: 1

Author

Michel Lagneau, Sep 15 2011

Keywords

• nonn

Comments

a(n) is of the form m^2 or 2*m^2.

See the comments in A195268 (numbers such that the sum of the odd divisors is prime).

It is interesting to observe that the intersection of this sequence with A195268 gives {9, 18, 36, 72, 144, 288, 576, 1152, 2304, 4608, 9216, 18432, 36864, 73728, 146689, 147456, 293378, 294912,...} and contains the sequence A005010(n) (numbers of the form 9*2^n), but is not equal to this sequence. For example, up to n = 400000, the numbers 146689 and 293378 are not divisible by 9.

Examples

			The divisors of 18 are  { 1, 2, 3, 6, 9, 18}, and the sum of the cube of the odd divisors 1^3 + 3^3 + 9^3 =757 is prime. Hence 18 is in the sequence.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A005010, A066100 (sqrt of odd numbers here), A195268.

Programs

Maple

with(numtheory):for n from 1 to 400000 do:x:=divisors(n):n1:=nops(x):s:=0:for m from 1 to n1 do:if irem(x[m],2)=1 then s:=s+x[m]^3:fi:od:if type(s,prime)=true  then printf(`%d, `,n): else fi:od:

Mathematica

  Module[{c=Range[800]^2,m},m=Sort[Join[c,2c]];Select[m,PrimeQ[Total[ Select[ Divisors[#],OddQ]^3]]&]](* Harvey P. Dale, Jul 31 2012 *)