A164928 Sum of the odd prime divisors of numbers whose odd prime divisors are all of the form 4k+3.

3, 3, 7, 3, 11, 3, 7, 3, 19, 10, 11, 23, 3, 3, 7, 31, 14, 3, 19, 10, 43, 11, 23, 47, 3, 7, 3, 7, 22, 59, 31, 10, 14, 67, 26, 71, 3, 19, 18, 79, 3, 83, 10, 43, 11, 23, 34, 47, 3, 7, 14, 103, 107, 3, 7, 22, 59, 11, 31, 10, 127, 46, 131, 14, 26, 67, 26, 139, 50, 71, 3, 10, 151, 19, 18

Offset: 1

Author

Jonathan Vos Post, Aug 31 2009

Keywords

• easy
• nonn

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.

Examples

			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.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000040, A002145, A009003, A164927-A164930.

Programs

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 *)

Extensions

Edited and extended by R. J. Mathar, Sep 08 2009