

A330724


Sum of prime factors (with multiplicity) of the nth composite number coprime to 6.


0



10, 12, 14, 16, 18, 18, 22, 20, 24, 28, 24, 22, 15, 26, 24, 34, 36, 30, 26, 17, 42, 28, 36, 46, 30, 48, 38, 30, 52, 19, 32, 34, 44, 58, 21, 48, 34, 64, 36, 50, 66, 40, 36, 23, 54, 72, 42, 21, 76, 38, 78, 60, 42, 23, 40, 84, 44, 48, 66, 88, 27, 68, 42, 94, 52, 25, 74
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OFFSET

1,1


COMMENTS

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.
From Cara Bennett, Dec 29 2019: (Start)
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.
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)


LINKS

Table of n, a(n) for n=1..67.


FORMULA

a(n) = A001414(A038509(n)).  Michel Marcus, Dec 28 2019


EXAMPLE

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.


MATHEMATICA

Map[Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#]] &, Select[Flatten[6 # + {1, 1} & /@ Range[90]], CompositeQ]] (* Michael De Vlieger, Dec 29 2019 *)
Total[Times@@@FactorInteger[#]]&/@Select[Range[500], CompositeQ[#] && CoprimeQ[ 6, #]&] (* Harvey P. Dale, Oct 24 2022 *)


CROSSREFS

Cf. A001414, A038509 (composite numbers of the form 6k + 1).
Sequence in context: A095406 A337028 A050769 * A351998 A088381 A163750
Adjacent sequences: A330721 A330722 A330723 * A330725 A330726 A330727


KEYWORD

nonn


AUTHOR

Nico Grimm, Dec 28 2019


STATUS

approved



