A164930 - OEIS

%I #8 Apr 06 2019 23:20:53

%S 5,5,11,17,5,11,23,5,29,17,5,41,11,23,47,5,53,16,29,59,17,71,5,41,83,

%T 22,11,89,23,47,5,101,53,107,16,113,28,29,59,11,5,131,17,137,71,34,

%U 149,5,41,83,167,22,173,11,89,179,23,28,47,191,197,5,101,46,53,107,16,113,227,28

%N Sum of odd prime divisors of numbers with all odd prime divisors of the form 6k+5.

%C We define a sequence b(n) = 5, 10, 11, 17, 20, 22, 23, 25, 29, 34, 40, 41, 44, 46, 47, 50, 53, 55, 58, ... to consist of those numbers where all odd prime factors are primes contained in A007528, and which have at least one prime factor in this class. 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.

%C Analogous sequence for primes of form 4k+1 is A164927.

%C Analogous sequence for primes of form 4k+3 is A164928.

%C Analogous sequence for primes of form 6k+1 is A164929.

%C The sum of an even number of primes of form 6k+1 is even (hence composite).

%C The sum of 3 primes of form 6n+5 is composite because (6a+5)+(6b+5)+(6c+5) = 3*(a+b+c+3).

%C However, the sum of 5 primes of form 6n+5 may be prime:

%C The smallest number, all of whose prime factors are of form 6k+5, whose sum of distinct prime factors is prime: 881705 = 5 * 11 * 17 * 23 * 41, and 5 + 11 + 17 + 23 + 41 = 97 is prime.

%e a(18) = 16 because b(18)= 55 = 5*11, and 5+11 = 16.

%p 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 6 <> 5 then RETURN(false) ; fi; od: RETURN(true) ; fi; end:

%p b := proc(n) if n = 1 then 5; else for a from procname(n-1)+1 do if isb(a) then RETURN(a) ; fi; od: fi; end:

%p A164930 := proc(n) local f; numtheory[factorset]( b(n)) minus {2} ; add(f,f=%) ; end: seq(A164930(n),n=1..120) ; # _R. J. Mathar_, Sep 09 2009

%Y Cf. A000040, A007528, A164927-A164930.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Aug 31 2009

%E Edited and extended by _R. J. Mathar_, Sep 09 2009