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This question is related to numbers found in the OEIS sequence A191837.

In this sequence, $a(2) = 48 = 5 + 7 + 17 + 19$, where the summands of 48 are all prime numbers that are less than or equal to $48/2=24$. The prime factors of 48 are $2^4$ and $3$, and neither $2$ nor $3$ are summands in $a(2)$. Similarly, $a(3) = 108 = 5 + 7 + 11 + 19 + 29 + 37$ and the prime factors of $108$ ($2^2$ and $3^3$) are not summands in $a(3)$. This appears to hold for all fourteen (verified and unverified) integers in the sequence.

Is there some way to prove or disprove the conjecture that for every integer in sequence A191837, the prime factors of $a(n)$ will never be an element of the summands of $a(n)$?

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    I have exte$n$ded this seque$n$ce by more that $2$$0$0 terms. Does anyone want to *try* to inde$p$endently verify my results? This was a challenging problem.2011-07-14

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Yes, of course, it's easy to prove the statement. In fact, the greatest common divisor of $a(n)$ and any of the summands $s$ in $a(n)$ has to be equal to one - they have to be relatively prime.

If they were not relatively prime, then we could look at the greatest common divisor $g>1$ of $s$ and $a(n)$. If $g, then $s$ wouldn't be a prime because it would have a divisor $g$, and such composite $s$ isn't allowed as a summand of $a(n)$ in the sequence.

If $g=s$, then we couldn't prove that $s$ is non-prime. However, $a(n)-s$ would also be a multiple of $g$ but we would have $a(n)-s > a(n)/2 > g$, so $a(n)-s$ wouldn't be a prime which would again violate the condition that both the summands $s$ and the differences $a(n)-s$ have to be prime for the elements $a(n)$ of your sequence (check the page you linked to carefully how the numbers are defined).

Not only you can omit candidate summands of $a(n)$ that are prime factors of $a(n)$: you may omit all candidate summands that fail to be relatively prime with $a(n)$. Well, truth to be told, it's almost the same thing if you know that the summands have to be prime.

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    Thanks Lubos! I just needed to be certain that removing the factors of m wouldn't cause erroneous results, and now I am.2011-07-09