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A330724 Sum of prime factors (with multiplicity) of the n-th 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 (list; graph; refs; listen; history; text; internal format)
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 *)

CROSSREFS

Cf. A001414, A038509 (composite numbers of the form 6k +- 1).

Sequence in context: A095406 A337028 A050769 * A088381 A163750 A167153

Adjacent sequences:  A330721 A330722 A330723 * A330725 A330726 A330727

KEYWORD

nonn

AUTHOR

Nico Grimm, Dec 28 2019

STATUS

approved

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Last modified July 27 15:49 EDT 2021. Contains 346308 sequences. (Running on oeis4.)