A084200 -id:A084200

Search: a084200 -id:a084200

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A082539

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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

FORMULA

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

MATHEMATICA

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

CROSSREFS

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)).

KEYWORD

nonn

AUTHOR

Benoit Cloitre, May 11 2003

EXTENSIONS

More terms from Reinhard Zumkeller, May 18 2003

STATUS

approved

A084196

Number of primes q<prime(n) such that q+1 divides prime(n)+1.

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

COMMENTS

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

LINKS

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

EXAMPLE

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

and 5, therefore a(5)=3.

MATHEMATICA

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

PROG

(Haskell)

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

CROSSREFS

Cf. A084198, A084200.

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, May 18 2003

STATUS

approved

A084197

Primes p such that there exists at least one prime q<p with q+1 dividing p+1.

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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

CROSSREFS

Cf. A082539, A084198, A084200.

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, May 18 2003

EXTENSIONS

Corrected by T. D. Noe, Oct 25 2006

STATUS

approved

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