A227680 Numbers whose sum of semiprime divisors is a prime number.

30, 36, 42, 66, 70, 72, 78, 105, 108, 114, 130, 144, 154, 165, 174, 182, 196, 210, 216, 222, 231, 238, 246, 255, 273, 282, 285, 286, 288, 310, 318, 324, 345, 357, 366, 370, 385, 392, 399, 418, 430, 432, 434, 441, 442, 455, 462, 465, 474, 483, 494, 498, 518

Offset: 1

Author

Michel Lagneau, Jul 19 2013

Keywords

• nonn

Comments

There exists a subsequence of infinite squares {36, 144, 196, 324, 441, 576, 676, 784, 1089, 1225, 1296, 1764,...} because the numbers of the form n = (p*q)^2 with p and q primes are in the sequence if p^2 + p*q + q^2 is prime (subsequence of A007645), and the numbers p^2, p*q and q^2 are the three possible semiprime divisors of n. This numbers of the sequence are 6^2, 14^2, 21^2, 26^2, 33^2, 35^2, 51^2, 69^2,...

The numbers of the form n = (p^a*q^v)^2 are also in the sequence => the sequence is infinite.

There exists a subsequence of numbers having three distinct prime divisors p, q and r such that p*q+q*r+r*p is prime (see A087054). This numbers are 30, 42, 66, 70, 78, 105, 114, ...

Examples

			30 is in the sequence because the semiprime divisors of 30 are 2*3, 2*5 and 3*5 and the sum 6+10+15 = 31 is a prime number.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007645 (primes of the form x^2 + xy + y^2).

Cf. A087054 (primes of the form p*q + q*r + r*p where p, q and r are distinct prime numbers).

Programs

Maple

with(numtheory):for n from 2 to 600 do:x:=divisors(n):n1:=nops(x): y:=factorset(n):n2:=nops(y):s1:=0:s2:=0:for i from 1 to n1 do: if bigomega(x[i])=2 then s1:=s1+x[i]:else fi:od: s2:=sum('y[i]', 'i'=1..n2):if type(s1,prime)=true then printf(`%d, `,n):else fi:od:

Mathematica

semipSigma[n_] := DivisorSum[n, # &, PrimeOmega[#] == 2 &]; Select[Range[500], PrimeQ @ semipSigma[#] &] (* Amiram Eldar, May 10 2020 *)