cp's OEIS Frontend

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

%I A227680 #13 May 10 2020 04:31:04
%S A227680 30,36,42,66,70,72,78,105,108,114,130,144,154,165,174,182,196,210,216,
%T A227680 222,231,238,246,255,273,282,285,286,288,310,318,324,345,357,366,370,
%U A227680 385,392,399,418,430,432,434,441,442,455,462,465,474,483,494,498,518
%N A227680 Numbers whose sum of semiprime divisors is a prime number.
%C A227680 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,...
%C A227680 The numbers of the form n = (p^a*q^v)^2 are also in the sequence => the sequence is infinite.
%C A227680 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, ...
%H A227680 Amiram Eldar, <a href="/A227680/b227680.txt">Table of n, a(n) for n = 1..10000</a>
%e A227680 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.
%p A227680 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:
%t A227680 semipSigma[n_] := DivisorSum[n, # &, PrimeOmega[#] == 2 &]; Select[Range[500], PrimeQ @ semipSigma[#] &] (* _Amiram Eldar_, May 10 2020 *)
%Y A227680 Cf. A007645 (primes of the form x^2 + xy + y^2).
%Y A227680 Cf. A087054 (primes of the form p*q + q*r + r*p where p, q and r are distinct prime numbers).
%K A227680 nonn
%O A227680 1,1
%A A227680 _Michel Lagneau_, Jul 19 2013