A078502 a(n) = smallest positive integer N such that (N - k)/k is prime for k = 1, 2, ..., n.

3, 6, 12, 12, 174600, 7224840, 10780560, 10780560, 1086338816640, 50060257410240, 7720634052774720, 227457297898150320, 7272877497848202240, 7272877497848202240

Offset: 1

Author

Joseph L. Pe, Jan 05 2003

Keywords

• nonn
• more

Comments

The idea for the sequence and first eleven terms are from Ken Wilke.

a(n) == 0 (mod 120) for n > 4: because a(n)/2, a(n)/3, a(n)/4 and a(n)/5 must be integer, a(n) == 0 (mod 60); and if a(n) == 60 (mod 120), (a(n)-4)/4 == 14 (mod 120) would not be prime; thus a(n) == 0 (mod 120). A more general result is a(n) == 0 (mod lcm(1,2,...,n)) for all n >= 1, and a(n) == 0 (mod 2*lcm(1,2,...,n)) for n > 4. - Jean-Christophe Hervé, Sep 15 2014

Examples

			(12-k)/k is prime for k = 1,2,3,4 and 12 is the smallest positive integer satisfying this property. Hence a(4) = 12.
		

Links

• Carlos Rivera, Puzzle 206. (N-k)/k Primes, Prime Puzzles and Problems Connection.
• Carlos Rivera, Puzzle 977. A special set of primes, Prime Puzzles and Problems Connection.
• G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 7272877497848202240

Crossrefs

See A093554 for another version.

Cf. A074200 (equivalent sequence for (N+k)/k prime).

Programs

PARI

a(n)=k=1; while(k,c=0; for(i=1,n,if(k%i==0&&isprime(k/i-1),c++)); if(c==n,return(k));k++)
n=1;while(n<10,print1(a(n),", ");n++) \\ Derek Orr, Sep 15 2014

Formula

a(n) == 0 (mod A003418(n)) because of the divisibility condition (A003418(n) = lcm(1,2,...,n)). - Jean-Christophe Hervé, Sep 15 2014

Extensions

Corrected and extended by Jens Kruse Andersen, Jan 10 2003