A271101 - OEIS

%I #14 Dec 15 2019 08:36:57

%S 21,33,57,69,85,93,129,133,145,177,205,213,217,237,249,253,265,309,

%T 393,417,445,469,489,493,505,517,553,565,573,597,633,669,685,697,753,

%U 781,793,813,817,865,889,913,933,949,973,985,993,1057,1077,1137,1149,1177,1257,1273,1285,1329

%N Squarefree semiprimes (A006881) whose average prime factor is prime.

%C Sum of factors of a(n) if semiprime (product 2*p, with p prime).

%C This sequence is subsequence of A006881, A089765, A187073, A108633 and A213015.

%C This sequence is also subsequence of A045835, because sopfr(omega(a(n))) = omega(sopfr(a(n))): sopfr(omega(a(n)))=sopfr(2)=2, and omega(sopfr(a(n)))=omega(2*p)=2  (p prime,  p>2, average prime factor).

%H Robert Israel, <a href="/A271101/b271101.txt">Table of n, a(n) for n = 1..10000</a>

%e 133 is in the sequence because 133 is a squarefree semiprime: 133=7*19, and (7+19)/2=13, a prime number.

%p N:= 10000: # for terms <= N

%p Primes:= select(isprime, [seq(i, i=3..N/3)]):

%p SP:= [seq(seq([p, q], q = select(`<=`, Primes, min(p-1, N/p))), p=Primes)]:

%p B:= select(t -> isprime((t[1]+t[2])/2), SP):

%p sort(map(t -> t[1]*t[2], B)); # _Robert Israel_, Dec 14 2019

%t Select[Select[Range@ 1330, SquareFreeQ@ # && PrimeOmega@ # == 2 &], PrimeQ@ Mean[First /@ FactorInteger@ #] &] (* _Michael De Vlieger_, Mar 30 2016 *)

%o (PARI)

%o sopf(n)= { local(f, s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); return(s) }

%o {for (n=6, 2*10^3,  if(bigomega(n)==2&&omega(n)==2, m=sopf(n)/2;if(m==truncate(m),if(isprime(m), print1(n, ", ")))))}

%Y Cf. A006881, A046315, A046388, A115585, A187073, A089765, A108633, A213015.

%K nonn

%O 1,1

%A _Antonio Roldán_, Mar 30 2016