A105399 -id:A105399 - cp's OEIS frontend

A134240 Numbers which occur in A105399 but not in A126990.

79, 127, 163, 331, 379, 397, 439, 487, 499, 541, 673, 701, 739, 757, 769, 787, 907, 937, 947, 967, 977, 991, 1009, 1103, 1123, 1163, 1187, 1213, 1249, 1367, 1399, 1523, 1549, 1567, 1579, 1597, 1753, 1801

Offset: 1

Artur Jasinski, Oct 15 2007

nonn

Cf. A105399, A134196, A134197, A126990.

A049591 Odd primes p such that p+2 is composite.

Original entry on oeis.org

7, 13, 19, 23, 31, 37, 43, 47, 53, 61, 67, 73, 79, 83, 89, 97, 103, 109, 113, 127, 131, 139, 151, 157, 163, 167, 173, 181, 193, 199, 211, 223, 229, 233, 241, 251, 257, 263, 271, 277, 283, 293, 307, 313, 317, 331, 337, 349, 353, 359, 367, 373, 379, 383, 389

Offset: 1

Labos Elemer

nonn

Primes p such that nextprime(p)-p >= 4.
Primes p such that p+2 divides (p-1)!.
Odd primes n such that n!*B(n+1) is an integer, where B(k) are the Bernoulli numbers. - Benoit Cloitre, Feb 06 2002
Sequence appears also to give all n > 1 such that there is no prime p satisfying the inequality n < p < n+tau(n)^2 where tau(n)=A000005(n). - Benoit Cloitre, Apr 13 2002
Conjecture: start from any initial value f(1) >= 2 and define f(n) to be the largest prime factor of f(1) +f(2) + ... +f(n-1); then f(n) = n/2 + O(log(n)) and there are infinitely many primes p such that f(2p)=p. Conjecture: current sequence gives primes satisfying f(2p)=p when f(1)=3. - Benoit Cloitre, Jun 04 2003
Numbers k such that 2((k-1)! + 1) + k is divisible by k(k+2). For 7 and 13, the respective quotients are also in the sequence. Are there any other such k? - Ivan N. Ianakiev, Aug 03 2019. The next values of k with respective quotients in this sequence are 103, 1531, 1637. - Amiram Eldar, Jun 08 2020
Numbers k such that 4((k-1)! + 1) == k^2 (mod k(k+2)). - Thomas Ordowski, May 09 2020

			13 is here because it is prime and 15 is composite. Also 15 divides 12!.

T. D. Noe, Table of n, a(n) for n = 1..1000
K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc., 44 (2007), 1-18.
Index entries for primes, gaps between

Cf. A067774, A105399.

[k:k in PrimesInInterval(3,400)| not IsPrime(k+2)]; // Marius A. Burtea, Aug 03 2019

d:=4; M:=1000; t0:=[]; for n from 1 to M do p:=ithprime(n); if nextprime(p) - p >= d then t0:=[op(t0),p]; fi; od: t0;

Select[Prime[Range[100]], NextPrime[#] -#>=4 &] (* G. C. Greubel, Aug 22 2019 *)

isok(p) = isprime(p) && (p % 2) && !isprime(p+2); \\ Michel Marcus, Feb 25 2014

[nth_prime(n) for n in (1..100) if (nth_prime(n+1) - nth_prime(n)) >= 4] # G. C. Greubel, Aug 22 2019

More terms from Benoit Cloitre, Jun 04 2003

Edited by Don Reble, Dec 20 2006

A126990 Largest prime preceding geometric mean of prime(n) and prime(n+2).

Original entry on oeis.org

3, 3, 7, 7, 13, 13, 19, 23, 23, 31, 31, 37, 43, 47, 47, 53, 61, 61, 67, 73, 73, 83, 89, 89, 97, 103, 103, 109, 113, 113, 131, 131, 139, 139, 151, 151, 157, 167, 167, 173, 181, 181, 193, 193, 199, 199, 211, 223, 229, 233, 233, 241, 241, 251, 257, 263, 271, 271, 277

Offset: 1

Artur Jasinski, Jan 01 2007

nonn

With duplicates removed, seems to be a subsequence of A105399 and A105792. - M. F. Hasler, Jun 14 2007

P. Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004.

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

Cf. A007917, A073273.

<< NumberTheory`NumberTheoryFunctions` a = {}; Do[AppendTo[a,PreviousPrime[Sqrt[(Prime[x])*(Prime[x + 2])]]], {x, 1, 100}]; a

A126990(n)={ n=sqrtint(prime(n)*prime(n+2)); if( 0==n%2, n--); while(!isprime(n), n-=2); n } /* then vector(50,n,A126990(n)) displays a list of values, M. F. Hasler, Jun 14 2007 */

a(n)= precprime(sqrtint(prime(n)*prime(n+2))); \\ Michel Marcus, Nov 07 2013

a(n) = A007917(A073273(n)). - Michel Marcus, Nov 07 2013

Edited by M. F. Hasler, Jun 14 2007

Definition changed so that offset is now 1 by Michel Marcus, Nov 07 2013

A105575 Largest primes < numbers of the form 6k (duplicates removed).

Original entry on oeis.org

5, 11, 17, 23, 29, 31, 41, 47, 53, 59, 61, 71, 73, 83, 89, 101, 107, 113, 131, 137, 139, 149, 151, 157, 167, 173, 179, 181, 191, 197, 199, 211, 227, 233, 239, 241, 251, 257, 263, 269, 271, 281, 283, 293, 311, 317, 331, 337, 347, 353, 359, 367, 373, 383, 389

Offset: 1

Giovanni Teofilatto, May 03 2005

			11 is in the sequence because 11 is the largest prime < 12 = 6*2.

Cf. A105399.

PrevPrim[n_] := Block[{k = n - 1}, While[ !PrimeQ[k], k-- ]; k]; Union[ Table[ PrevPrim[6n], {n, 65}]] (* Robert G. Wilson v, May 21 2005 *)
NextPrime[#,-1]&/@(6*Range[100])//Union (* Harvey P. Dale, Sep 23 2017 *)

Corrected and extended by Robert G. Wilson v, May 21 2005

Edited, corrected and extended by Ray Chandler, Oct 17 2006

A105792 Largest prime <= numbers congruent (2,4) mod 6 (duplicates removed).

Original entry on oeis.org

2, 3, 7, 13, 19, 23, 31, 37, 43, 47, 53, 61, 67, 73, 79, 83, 89, 97, 103, 109, 113, 127, 131, 139, 151, 157, 163, 167, 173, 181, 193, 199, 211, 223, 229, 233, 241, 251, 257, 263, 271, 277, 283, 293, 307, 313, 317, 331, 337, 349, 353, 359, 367, 373, 379, 383, 389

Offset: 1

Giovanni Teofilatto, May 05 2005

Except for the first terms, largest prime p is at least < two numbers congruent (2,4) mod 6.

			7 is in the sequence because 7 is the largest prime < 8 which is a number congruent (2,4) mod 6.

Cf. A105399, A105575.

pp[n_] := Block[{k = n},While[ ! PrimeQ[k], k-- ];k];Union[pp /@ Select[Range[400], MemberQ[{2, 4}, Mod[ #, 6]] &]] (* Ray Chandler, Oct 17 2006 *)
Union[Abs[NextPrime[#,-1]&/@Select[Range[400],MemberQ[{2,4}, Mod[ #,6]]&]]] (* Harvey P. Dale, May 17 2012 *)

Extended by Ray Chandler, Oct 17 2006

cp's OEIS Frontend

A134240 Numbers which occur in A105399 but not in A126990.

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A049591 Odd primes p such that p+2 is composite.

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A126990 Largest prime preceding geometric mean of prime(n) and prime(n+2).

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A105575 Largest primes < numbers of the form 6k (duplicates removed).

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A105792 Largest prime <= numbers congruent (2,4) mod 6 (duplicates removed).

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