A093554 -id:A093554 - cp's OEIS frontend

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

Joseph L. Pe, Jan 05 2003

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

			(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.

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

See A093554 for another version.

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

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

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

Corrected and extended by Jens Kruse Andersen, Jan 10 2003

A181680 Primes p such that (p-1)/2, (p-2)/3, (p-3)/4, (p-4)/5 and (p-5)/6 are prime numbers.

Original entry on oeis.org

7224839, 9339119, 10780559, 42464519, 45652319, 53464319, 62198639, 69899759, 94159799, 121136399, 130777919, 158608799, 191010959, 285785639, 288308159, 322461719, 353256119, 402411239, 423803519

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 17 2010

nonn

Cf. A093554.

Select[Prime[Range[20000000]],PrimeQ[(#-1)/2]&&PrimeQ[(#-2)/3]&&PrimeQ[(#-3)/4]&&PrimeQ[(#-4)/5]&&PrimeQ[(#-5)/6]&]
Select[Prime[Range[2254*10^4]],AllTrue[(#-Range[5])/Range[2,6],PrimeQ]&] (* Harvey P. Dale, May 29 2024 *)

Edited, terms checked and two terms added by Klaus Brockhaus, Nov 17 2010

cp's OEIS Frontend

A168453 Erroneous version of A093554.

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

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A181680 Primes p such that (p-1)/2, (p-2)/3, (p-3)/4, (p-4)/5 and (p-5)/6 are prime numbers.

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