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)
FORMULA
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
KEYWORD
nonn
AUTHOR
Nico Grimm, Dec 28 2019
STATUS
approved