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%I #35 Oct 24 2022 13:51:27
%S 10,12,14,16,18,18,22,20,24,28,24,22,15,26,24,34,36,30,26,17,42,28,36,
%T 46,30,48,38,30,52,19,32,34,44,58,21,48,34,64,36,50,66,40,36,23,54,72,
%U 42,21,76,38,78,60,42,23,40,84,44,48,66,88,27,68,42,94,52,25,74
%N Sum of prime factors (with multiplicity) of the n-th composite number coprime to 6.
%C It appears that odd numbers appear rarely in this sequence. I conjecture that all the odd numbers above 15 and all the even numbers above 10 will at least once show up in the sequence.
%C From _Cara Bennett_, Dec 29 2019: (Start)
%C Every combination of prime numbers >= 5 when summed will be in this sequence (e.g., 5 + 5 = 10, 5 + 7 = 12, 7 + 7 = 14, ...). As long as the numbers 15 through 19 can be written as the sum of 2 or more prime numbers >= 5, then we can write any integer >= 15 in this form simply by adding fives to either 15, 16, 17, 18, or 19. Note that we can do this: 5 + 5 + 5 = 15; 5 + 11 = 16; 5 + 5 + 7 = 17; 5 + 13 = 18; 5 + 7 + 7 = 19.
%C For example, if we wanted to show that 330724 will appear in the sequence, we can write 330724 as the sum 330705 + 19 = (5 + 5 + ... + 5) + 5 + 7 + 7. The product (5 * 5 * ... * 5) * (5 * 7 * 7) is a composite number coprime to 6. Hence, 330724 must be in the sequence. (End)
%F a(n) = A001414(A038509(n)). - _Michel Marcus_, Dec 28 2019
%e The first composite number in the referenced sequence is 25. The prime factorization of 25 is 5^2 or 5*5. The sum of these prime factors is 10. Therefore a(1) = 10.
%t Map[Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#]] &, Select[Flatten[6 # + {-1, 1} & /@ Range[90]], CompositeQ]] (* _Michael De Vlieger_, Dec 29 2019 *)
%t Total[Times@@@FactorInteger[#]]&/@Select[Range[500],CompositeQ[#] && CoprimeQ[ 6,#]&] (* _Harvey P. Dale_, Oct 24 2022 *)
%Y Cf. A001414, A038509 (composite numbers of the form 6k +- 1).
%K nonn
%O 1,1
%A _Nico Grimm_, Dec 28 2019
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