OFFSET
1,1
COMMENTS
We define a sequence b(n) = 3, 6, 7, 9, 11, 12, 14, 18, 19, 21, 22, 23, ... to consist of those numbers where all odd prime factors are primes contained in A002145, and which have at least one prime factor in this class; b(n) is basically A004144 without the powers of 2.
a(n) is the sum of the distinct odd prime factors of b(n), where "distinct" means that the multiplicity (exponent) in the prime factorization of b(n) is ignored.
Analogous sequence for primes of form 4k+1 is A164927.
Analogous sequence for primes of form 6k+1 is A164929.
Analogous sequence for primes of form 6k+5 is A164930.
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..1000
EXAMPLE
a(11) = 10 because b(11) = 21 = 3*7, and 3+7 = 10.
The smallest nonprime number, all of whose prime factors are of form 4n+3, whose sum of distinct prime factors is prime: b(181) = 3*7*19 = 399; 3+7+19 = 29.
MAPLE
isb := proc(n) fs := numtheory[factorset](n) minus {2} ; if fs = {} then RETURN(false); else for f in fs do if op(1, f) mod 4 <> 3 then RETURN(false) ; fi; od: RETURN(true) ; fi; end:
b := proc(n) if n = 1 then 3; else for a from procname(n-1)+1 do if isb(a) then RETURN(a) ; fi; od: fi; end:
A164928 := proc(n) local f; numtheory[factorset]( b(n)) minus {2} ; add(f, f=%) ; end: seq(A164928(n), n=1..120) ; # R. J. Mathar, Sep 08 2009
MATHEMATICA
sopd[n_]:=Module[{ff=Select[Transpose[FactorInteger[n]][[1]], OddQ]}, If[ And@@ (Mod[#, 4]==3&/@ff), Total[ff], 0]]; Select[Array[sopd, 200], #>0&] (* Harvey P. Dale, Dec 16 2013 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Aug 31 2009
EXTENSIONS
Edited and extended by R. J. Mathar, Sep 08 2009
STATUS
approved