A164865 Sum of the distinct semiprime divisors of the n-th number with two or more distinct semiprime divisors.

10, 15, 14, 10, 18, 31, 19, 14, 41, 26, 24, 10, 35, 30, 15, 18, 35, 30, 61, 38, 59, 19, 40, 42, 71, 14, 45, 26, 40, 50, 10, 63, 42, 39, 91, 30, 71, 19, 87, 18, 101, 62, 48, 35, 66, 50, 101, 65, 24, 38, 121, 63, 19, 70, 78, 56, 42, 60, 113, 75, 14, 15, 86, 103, 45, 129, 66, 90

Offset: 1

Author

Jonathan Vos Post, Aug 28 2009

Keywords

• easy
• nonn

Comments

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

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

Examples

			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.
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.
		

Crossrefs

Cf. A001358, A068318, A086971, A102467.

Programs

Maple

isA001358 := proc(n) RETURN( numtheory[bigomega](n) =2 ) ; end:
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:
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:
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:
A164865 := proc(n) A076290( A102467(n+1)) ; end: seq(A164865(n),n=1..120) ; # R. J. Mathar, Aug 31 2009

Mathematica

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 *)

Formula

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

a(n) = A076290( A102467(n+1)). - R. J. Mathar, Aug 31 2009

Extensions

Corrected and extended by R. J. Mathar, Aug 31 2009