A114985 - OEIS

%I #11 Sep 14 2015 10:36:01

%S 14,21,26,28,30,33,38,46,52,56,57,60,62,63,69,70,74,76,85,90,92,93,94,

%T 98,99,102,104,105,106,112,120,124,129,133,134,140,145,147,148,150,

%U 152,166,171,174,177,178,180,182,184,188,189,190,195,196,204,205,207,208

%N Numbers whose sum of distinct prime factors is semiprime.

%C This is the semiprime analog of A114522 "numbers n such that sum of distinct prime divisors of n is prime." See also A110893 "numbers with a semiprime number of prime divisors (counted with multiplicity)."

%F {k such that A008472(k) is an element of A001358}. {k such that sopf(k) is an element of A001358}. {k = Product(Prime(j)^e_j) such that Sum(Prime(j)) is in A001358}.

%e a(1) = 14 because 14 = 2 * 7 and 2 + 7 = 9 = 3^2 is semiprime.

%e a(2) = 21 because 21 = 3 * 7 and 3 + 7 = 10 = 2 * 5 is semiprime.

%e a(3) = 26 because 26 = 2 * 13 and 2 + 13 = 15 = 3 * 5 is semiprime.

%e a(4) = 28 because 28 = 2^2 * 7 and 2 + 7 = 9 = 3^2 is semiprime.

%e a(5) = 30 because 30 = 2 * 3 * 5 and 2 + 3 + 5 = 10 = 2 * 5 is semiprime.

%e a(6) = 33 because 33 = 3 * 11 and 3 + 11 = 14 = 2 * 7 is semiprime.

%e a(7) = 38 because 38 = 2 * 19 and 2 + 19 = 21 = 3 * 7 is semiprime.

%p with(numtheory): a:=proc(n) local A,s,B: A:=factorset(n): s:=sum(A[j],j=1..nops(A)): B:=factorset(s): if nops(B)=2 and B[1]*B[2]=s or nops(B)=1 and B[1]^2=s then n else fi end: seq(a(n),n=2..250); # _Emeric Deutsch_, Mar 07 2006

%t Select[Range[250],PrimeOmega[Total[Transpose[FactorInteger[#]][[1]]]]==2&] (* _Harvey P. Dale_, May 06 2013 *)

%o (PARI) is(n)=bigomega(vecsum(factor(n)[,1]))==2 \\ _Charles R Greathouse IV_, Sep 14 2015

%Y Cf. A001358, A008472, A110893, A114522.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Feb 22 2006

%E Corrected and extended by _Emeric Deutsch_, Mar 07 2006