A242932 Numbers n such that k*n/(k+n) is prime for some k.

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

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 (without loss of generality, 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: A135667 A156624 A341340 * A351863 A025201 A071259

Adjacent sequences: A242929 A242930 A242931 * A242933 A242934 A242935

Keyword

nonn

Author

Derek Orr, May 27 2014

Status

approved