login
A164865
Sum of the distinct semiprime divisors of the n-th number with two or more distinct semiprime divisors.
2
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
1,1
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).
FORMULA
a(n) = Sum_(k|A102467(n) and k in A001358) k.
a(n) = A076290( A102467(n+1)). - R. J. Mathar, Aug 31 2009
EXAMPLE
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.
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 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Aug 28 2009
EXTENSIONS
Corrected and extended by R. J. Mathar, Aug 31 2009
STATUS
approved