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A242932 Numbers n such that k*n/(k+n) is prime for some k. 1
3, 4, 6, 8, 10, 12, 14, 18, 20, 22, 24, 26, 30, 32, 34, 38, 42, 44, 46, 48, 54, 56, 58, 60, 62, 68, 72, 74, 80, 82, 84, 86, 90, 94, 98, 102, 104, 106, 108, 110, 114, 118, 122, 128, 132, 134, 138, 140, 142, 146, 150, 152, 158, 164, 166, 168, 174, 178, 180, 182, 192, 194 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A subsequence of a(n) is the prime numbers plus 1 (A008864).

a(n) is even for all n > 1. Proof: There are four possibilities for n and k: odd-odd, odd-even, even-even (WLOG even-odd and odd-even are the same). If k and n are odd, then the numerator is odd and the denominator is even. Thus, this will never be an integer or prime. If k and n are even, the numerator is even and the denominator is even. An even divided by an even could be odd or even so primes are a possibility. If one is odd and one is even, the numerator is even and the denominator is odd. The only way this is prime is if it equals 2. Thus, letting k = 2a and n = 2b+1, then 2a*(2b+1)/(2a+2b+1) = 2. Solving this, we get that a=3 and b=1 (meaning k = 6 and n = 3). So, 3 is the only odd number in this sequence.

It is believed that numbers in A016742 (except 4) are not included in this sequence.

LINKS

Table of n, a(n) for n=1..62.

EXAMPLE

4*k/(4+k) is prime for some k (let k = 4).

PROG

(PARI) a(n)=for(k=1, n*(n-1), s=(k*n)/(k+n); if(floor(s)==s, if(ispseudoprime(s), return(k))))

n=1; while(n<1000, if(a(n), print1(n, ", ")); n+=1)

CROSSREFS

Cf. A008864, A016742.

Sequence in context: A204662 A135667 A156624 * A025201 A071259 A231405

Adjacent sequences:  A242929 A242930 A242931 * A242933 A242934 A242935

KEYWORD

nonn

AUTHOR

Derek Orr, May 27 2014

STATUS

approved

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Last modified December 16 04:05 EST 2019. Contains 330013 sequences. (Running on oeis4.)