This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A224830 #23 May 10 2020 02:43:18
%S A224830 36,72,108,144,165,210,216,273,288,324,345,385,399,432,462,561,576,
%T A224830 595,648,651,665,715,795,798,858,864,885,957,972,1001,1015,1110,1152,
%U A224830 1218,1281,1290,1296,1335,1443,1463,1495,1515,1533,1547,1551,1615,1645,1659
%N A224830 Numbers n such that both the sum of the semiprime divisors of n and the sum of the prime divisors of n are prime numbers.
%C A224830 Numbers n such that A008472(n) and A076290(n) are both prime numbers.
%C A224830 There exists a subsequence of squares {36, 144, 324, 576, 1296, 2304, 2916, 5184, 9216, 11664, 20736, 26244, 36864, ...} and the numbers of the form n = (p*q)^2 or (p^a*q^v)^2 with p and q primes are in the sequence if we have the two conditions:
%C A224830 (1) p+q = p1 is prime => p=2
%C A224830 (2) p^2 + p*q + q^2 = p2 is prime (subsequence of A007645), because p^2, p*q and q^2 are the three possible semiprime divisors of n, but with p=2, the semiprime divisors are 4, 2q and q^2.
%C A224830 (1) and (2) => p2 - 2*p1 = q^2, hence the property:
%C A224830 Let a number n such that the sum of the semiprime divisors is a prime number p1 and the sum of the prime divisors of n is a prime number p2. If n is a perfect square having two prime divisors, then p1 - 2*p2 = 9. Proof:
%C A224830 If q > 3, q == 1 mod 6 => q^2 + 2q + 4 == 1 mod 6 (if q==5 mod 6, q^2 + 2q + 4 == 3 mod 6 is not prime), but q+2 == 3 mod 6 is not prime. Conclusion: q = 3, and q^2 = 9 if a(n) is a square.
%C A224830 Consequence: if a(n) is a square having two prime divisors, the number k*a(n) with k = 2 or 3 is in the sequence.
%H A224830 Amiram Eldar, <a href="/A224830/b224830.txt">Table of n, a(n) for n = 1..10000</a>
%e A224830 72 is in the sequence because the sum of the prime divisors is 2+3 = 5 and the sum of the semiprime divisors is 4 + 2*3 + 9 = 19.
%p A224830 with(numtheory):for n from 2 to 2000 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 and type(s2,prime)=true then printf(`%d, `,n):else fi:od:
%t A224830 primeSum[n_] := Plus @@ First[Transpose[FactorInteger[n]]]; semipSigma[n_] := DivisorSum[n, # &, PrimeOmega[#] == 2 &]; Select[Range[2000], PrimeQ @ primeSum[#] && PrimeQ @ semipSigma[#] &] (* _Amiram Eldar_, May 10 2020 *)
%Y A224830 Intersection of A114522 and A227680.
%Y A224830 Cf. A008472, A076290.
%K A224830 nonn
%O A224830 1,1
%A A224830 _Michel Lagneau_, Jul 21 2013