A084200 -id:A084200 - cp's OEIS frontend

A082539 Primes p such that there is no prime q, q < p with q+1 dividing p+1.

2, 3, 13, 37, 61, 73, 109, 157, 193, 229, 241, 277, 313, 337, 373, 397, 409, 421, 457, 541, 577, 613, 661, 673, 709, 733, 757, 829, 877, 997, 1009, 1033, 1069, 1093, 1117, 1129, 1153, 1201, 1213, 1237, 1249, 1297, 1321, 1381, 1453, 1489, 1597, 1621, 1657

Offset: 1

Benoit Cloitre, May 11 2003

nonn

Contains A005383, primes p such that (p+1)/2 is prime. - T. D. Noe, Apr 28 2004

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A005383, A049084, A084196, A084198, A084200.

Cf. A092307 (primes p such that there are no primes q, 3 < q < p, such that (q-1) divides (p-1)).

Select[Prime[Range[260]], AllTrue[Most[Divisors[# + 1]], !PrimeQ[#1 - 1] &] &] (* Amiram Eldar, Jun 06 2022 *)

A084196(A049084(a(n))) = 0.

More terms from Reinhard Zumkeller, May 18 2003

A084196 Number of primes q

Original entry on oeis.org

0, 0, 1, 1, 3, 0, 2, 1, 5, 2, 2, 0, 3, 1, 6, 3, 6, 0, 1, 7, 0, 3, 6, 4, 1, 2, 2, 6, 0, 3, 3, 5, 2, 3, 3, 3, 0, 1, 9, 2, 9, 1, 8, 0, 3, 3, 1, 4, 6, 0, 3, 11, 0, 8, 2, 8, 6, 3, 0, 2, 1, 5, 3, 7, 0, 2, 1, 0, 5, 1, 2, 13, 2, 0, 3, 10, 3, 0, 2, 0, 11, 0, 11, 2, 5, 5, 6, 0, 4, 2, 6, 13, 2, 5, 2, 13, 4, 4, 1, 0, 1, 4

Offset: 1

Reinhard Zumkeller, May 18 2003

nonn

a(A049084(A082539(n)))=0, a(A049084(A084197(n)))>0, a(A049084(A084198(n)))=1;

			n=5, prime(5)=11: (11+1) mod (q+1) = 0 for 3 primes q<11: 2, 3,
and 5, therefore a(5)=3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A084198, A084200.

a084196 n = a084196_list !! (n-1)
a084196_list = f [] a000040_list where
   f ps' (p:ps) = length [q | q <- ps', mod (p + 1) (q + 1) == 0] :
                  f (p : ps') ps where
-- Reinhard Zumkeller, Jan 06 2014

Table[Count[Mod[p+1,Prime[Range[PrimePi[p]-1]]+1],0],{p,Prime[Range[110]]}] (* Harvey P. Dale, Aug 11 2023 *)

A084197 Primes p such that there exists at least one prime q

Original entry on oeis.org

5, 7, 11, 17, 19, 23, 29, 31, 41, 43, 47, 53, 59, 67, 71, 79, 83, 89, 97, 101, 103, 107, 113, 127, 131, 137, 139, 149, 151, 163, 167, 173, 179, 181, 191, 197, 199, 211, 223, 227, 233, 239, 251, 257, 263, 269, 271, 281, 283, 293, 307, 311, 317, 331, 347, 349, 353

Offset: 1

Reinhard Zumkeller, May 18 2003

nonn

A084196(A049084(a(n)))>0.

Matthew House, Table of n, a(n) for n = 1..10000

Cf. A082539, A084198, A084200.

Corrected by T. D. Noe, Oct 25 2006

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A082539 Primes p such that there is no prime q, q < p with q+1 dividing p+1.

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A084196 Number of primes q

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A084197 Primes p such that there exists at least one prime q

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