A164929 Sum of odd prime divisors of numbers with all odd prime divisors of the form 6k+1.

7, 13, 7, 19, 13, 7, 31, 37, 19, 43, 7, 13, 7, 61, 31, 67, 73, 37, 19, 79, 43, 20, 97, 7, 103, 13, 109, 7, 61, 31, 127, 26, 67, 139, 73, 37, 151, 19, 157, 79, 163, 13, 43, 181, 20, 193, 97, 7, 199, 103, 13, 211, 38, 109, 223, 7, 229, 241, 61, 32, 31, 127, 44, 26, 67, 271, 277

Offset: 1

Author

Jonathan Vos Post, Aug 31 2009

Keywords

• easy
• nonn

Comments

We define a sequence b(n) = 7, 13, 14, 19, 26, 28, 31, 37, 38, 43, 49, 52, ... to consist of those numbers where all odd prime factors are primes contained in A002476, and which have at least one prime factor in this class; b(n) is basically elements of A004611 multiplied by 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 4k+3 is A164928.

Analogous sequence for primes of form 6k+5 is A164930.

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

The sum of 3 primes of form 6k+1 is composite because (6a+1)+(6b+1)+(6c+1) = 3*(a+b+c+1). However (see 2nd example) the sum of 5 primes of form 6k+1 may be prime.

Examples

			a(22) = 20 because b(22) = 91 = 7*13, and 7+13 = 20.
The smallest number, all of whose prime factors are of form 6k+1, whose sum of distinct prime factors is prime: 1983163 = 7 * 13 * 19 * 31 * 37, and 7 + 13 + 19 + 31 + 37 = 107 is prime.
		

Crossrefs

Cf. A000040, A002476, 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 6 <> 1 then RETURN(false) ; fi; od: RETURN(true) ; fi; end:
b := proc(n) if n = 1 then 7; else for a from procname(n-1)+1 do if isb(a) then RETURN(a) ; fi; od: fi; end:
A164929 := proc(n) local f; numtheory[factorset]( b(n)) minus {2} ; add(f,f=%) ; end: seq(A164929(n),n=1..120) ; # R. J. Mathar, Sep 09 2009

Extensions

Edited and extended by R. J. Mathar, Sep 09 2009