A164865 - OEIS

%I #9 Mar 25 2019 02:31:35

%S 10,15,14,10,18,31,19,14,41,26,24,10,35,30,15,18,35,30,61,38,59,19,40,

%T 42,71,14,45,26,40,50,10,63,42,39,91,30,71,19,87,18,101,62,48,35,66,

%U 50,101,65,24,38,121,63,19,70,78,56,42,60,113,75,14,15,86,103,45,129,66,90

%N Sum of the distinct semiprime divisors of the n-th number with two or more distinct semiprime divisors.

%C The sum of semiprime divisors of all k such that A086971(k) > 1.

%C This sum is prime for k = 30, 36, 60, 72, and infinitely more values (every prime power of every primitive element).

%F a(n) = Sum_(k|A102467(n) and k in A001358) k.

%F a(n) = A076290( A102467(n+1)). - _R. J. Mathar_, Aug 31 2009

%e a(1) = 10 because the 1st number with 2 or more distinct semiprime divisors is k=12=A102467(2), as A001358(1) = 4, 4|12, A001358(2) = 6, 6|12, and 4+6 = 10.

%e a(6) = 31 because the 6th number with multiple distinct semiprime factors is k=30=A102467(7), the semiprimes 6, 10, and 15 divide 30, and 6 + 10 + 15 = 31.

%p isA001358 := proc(n) RETURN( numtheory[bigomega](n) =2 ) ; end:

%p A086971 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do if isA001358(d) then a := a+1; fi; od; a ; end:

%p A102467 := proc(n) local a; if n = 1 then 1; else for a from procname(n-1)+1 do if A086971(a) >= 2 then RETURN(a) ; fi; od: fi; end:

%p A076290 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do if isA001358(d) then a := a+d; fi; od; a ; end:

%p A164865 := proc(n) A076290( A102467(n+1)) ; end: seq(A164865(n),n=1..120) ; # _R. J. Mathar_, Aug 31 2009

%t sdsd[n_]:=Module[{spd=Select[Divisors[n],PrimeOmega[#]==2&]}, If[ Length[ spd]> 1,Total[spd],0]]; DeleteCases[Array[sdsd,200],0] (* _Harvey P. Dale_, Oct 29 2015 *)

%Y Cf. A001358, A068318, A086971, A102467.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Aug 28 2009

%E Corrected and extended by _R. J. Mathar_, Aug 31 2009