A084196 -id:A084196 - 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

A084198 Primes p such that there exists exactly one prime q

Original entry on oeis.org

5, 7, 19, 43, 67, 97, 163, 181, 211, 283, 331, 349, 523, 547, 601, 691, 787, 853, 907, 937, 1021, 1051, 1123, 1171, 1429, 1531, 1549, 1723, 1777, 1867, 2011, 2029, 2053, 2083, 2113, 2251, 2281, 2347, 2371, 2467, 2521, 2707, 2731, 2749, 2803, 2971, 3067

Offset: 1

Reinhard Zumkeller, May 18 2003

nonn

A084196(A049084(a(n)))=1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

a084198 n = a084198_list !! (n-1)
a084198_list = map a000040 $ filter ((== 1) . a084196) [1..]
-- Reinhard Zumkeller, Jan 06 2014

pdQ[n_]:=Module[{prs=Prime[Range[500]],p1},p1=Select[prs,#?(Divisible[n+1,#]&)]==1]; Select[Prime[Range[500]],pdQ] (* _Harvey P. Dale, Dec 24 2013 *)

Corrected by T. D. Noe, Oct 25 2006

A084200 LeastCommonMultiple{q+1: q prime, q < prime(n), q+1 divides prime(n)+1}.

Original entry on oeis.org

1, 1, 3, 4, 12, 1, 6, 4, 24, 6, 8, 1, 42, 4, 24, 18, 60, 1, 4, 72, 1, 40, 84, 90, 14, 6, 8, 108, 1, 114, 32, 132, 6, 140, 30, 152, 1, 4, 168, 6, 180, 14, 96, 1, 18, 40, 4, 224, 228, 1, 18, 240, 1, 252, 6, 264, 270, 136, 1, 6, 4, 294, 308, 312, 1, 6, 4, 1, 348, 14, 6, 360, 8, 1

Offset: 1

Reinhard Zumkeller, May 18 2003

nonn

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

Cf. A084196, A082539, A084197.

from math import lcm
from sympy import sieve
def a(n):
    pn = sieve[n]
    return lcm(*[q+1 for q in sieve.primerange(2, pn) if (pn+1)%(q+1)==0])
print([a(n) for n in range(1, 100)]) # Michael S. Branicky, Aug 13 2024

Definition and terms corrected by Matthew House, Aug 13 2024

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

A084199 Smallest prime q < prime(n) such that q+1 divides prime(n)+1, a(n) = 1 if no such q exists.

Original entry on oeis.org

1, 1, 2, 3, 2, 1, 2, 3, 2, 2, 3, 1, 2, 3, 2, 2, 2, 1, 3, 2, 1, 3, 2, 2, 13, 2, 3, 2, 1, 2, 3, 2, 2, 3, 2, 3, 1, 3, 2, 2, 2, 13, 2, 1, 2, 3, 3, 3, 2, 1, 2, 2, 1, 2, 2, 2, 2, 3, 1, 2, 3, 2, 3, 2, 1, 2, 3, 1, 2, 13, 2, 2, 3, 1, 3, 2, 2, 1, 2, 1, 2, 1, 2, 13, 3, 2, 2, 1, 2, 3, 2, 2, 3, 2, 3, 2, 2, 2, 3, 1, 3, 2, 2

Offset: 1

Reinhard Zumkeller, May 18 2003

nonn

Conjecture: A008578 is the range of values (or A000040 = range of values > 1).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A008578, A084196.

seq[lim_] := Module[{p = Prime[Range[lim]], q, s = {}}, Do[q = SelectFirst[p, Divisible[p[[i]] + 1, # + 1] &]; AppendTo[s, If[q < p[[i]], q, 1]], {i, 1, Length[p]}]; s]; seq[100] (* Amiram Eldar, Mar 28 2025 *)

a(n) = 1 iff A084196(n) = 0.

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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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A084198 Primes p such that there exists exactly one prime q

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A084200 LeastCommonMultiple{q+1: q prime, q < prime(n), q+1 divides prime(n)+1}.

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

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A084199 Smallest prime q < prime(n) such that q+1 divides prime(n)+1, a(n) = 1 if no such q exists.

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