A068497 -id:A068497 - cp's OEIS frontend

A109998 Non-Cunningham primes: primes isolated from any Cunningham chain under any iteration of 2p+-1 or (p+-1)/2.

17, 43, 67, 71, 101, 103, 109, 127, 137, 149, 151, 163, 181, 197, 223, 241, 257, 269, 283, 311, 317, 349, 353, 373, 389, 401, 409, 433, 449, 461, 463, 487, 521, 523, 557, 569, 571, 599, 617, 631, 643, 647, 677, 701, 709, 739, 751, 769, 773, 787, 797, 821

Offset: 1

Alexandre Wajnberg, Sep 01 2005

The condition that neither 2p - 1 nor 2p + 1 be prime is equivalent to ((p-1) mod 3 = 0) or ((p+1) mod 3 = 0). For example, the prime p = 2^607 - 1 is not in this sequence because p + 1 mod 3 = 2. - Washington Bomfim, Oct 30 2009

			a(1) = 17 is here because 17 * 2 + 1 = 35, 17 * 2 - 1 = 33; (17+1)/2 = 9, (17-1)/2 = 8: four composite numbers.

Chris Caldwell's Prime Glossary, Cunningham chains.
Douglas S. Stones, On prime chains, arXiv:0908.2166 [math.NT] [From _Washington Bomfim_, Oct 30 2009, edited by _R. J. Mathar_, Mar 01 2010]

Cf. A005385, A005383, A060254, A005384, A005382, A068497, A059455, A059762, A057326, A023272, A023302, A059764, A057328, A023330, A005602, A064812, A005603.

nonCunninghamPrimes = {}; Do[p = Prime[n]; If[!PrimeQ[2p - 1] && !PrimeQ[2p + 1] && !PrimeQ[(p - 1)/2] && !PrimeQ[(p + 1)/2], AppendTo[nonCunninghamPrimes, p]], {n, 6!}]; nonCunninghamPrimes (* Vladimir Joseph Stephan Orlovsky, Mar 22 2009 *)

Corrected and extended by Ray Chandler, Sep 02 2005

Replaced link to cached arXiv URL with link to the abstract - R. J. Mathar, Mar 01 2010

A195685 Primes p for which tau(2p-1) = tau(2p+1) = 4.

Original entry on oeis.org

17, 43, 47, 71, 101, 107, 109, 151, 197, 223, 317, 349, 461, 521, 569, 631, 673, 701, 821, 881, 919, 947, 971, 991, 1051, 1091, 1109, 1153, 1181, 1217, 1231, 1259, 1321, 1361, 1367, 1549, 1693, 1801, 1847, 1933, 1951, 1979, 2143, 2207, 2267, 2297, 2441, 2801

Offset: 1

Timothy L. Tiffin, Sep 22 2011

nonn

Sequence terms are a subset of those listed in A086006 and A068497.

The numbers 2p-1, 2p, 2p+1 form a run (indeed, a maximal run) of three consecutive integers each with four positive divisors. The first two examples are 33, 34, 35 and 85, 86, 87. A039833 gives the first number in these maximal 3-integer runs. - Timothy L. Tiffin, Jul 05 2016

			tau(2*17-1) = tau(33) = tau(3*11) = 4 = tau(5*7) = tau(35) = tau(2*17+1) and tau(2*43-1) = tau(85) = tau(5*17) = 4 = tau(3*29) = tau(87) = tau(2*43+1). - _Timothy L. Tiffin_, Jul 05 2016

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A039833, A068497, A086006, A248201.

with(numtheory):
q:= p-> isprime(p) and tau(2*p-1)=4 and tau(2*p+1)=4:
select(q, [$1..3000])[];  # Alois P. Heinz, Apr 18 2019

Select[Prime[Range[500]], DivisorSigma[0, 2 # - 1] == DivisorSigma[0, 2 # + 1] == 4 &] (* T. D. Noe, Sep 22 2011 *)
Select[Mean[#]/2&/@SequencePosition[DivisorSigma[0,Range[6000]],{4,,4}],PrimeQ] (* _Harvey P. Dale, Nov 26 2021 *)

lista(nn) = forprime(p=2, nn, if ((numdiv(2*p-1) == 4) && (numdiv(2*p+1) == 4), print1(p, ", "))); \\ Michel Marcus, Jul 06 2016

a(n) = A248201(n)/2. - Torlach Rush, Jun 25 2021

A109927 First primes p connected to two primes either by 2p+1 or 2p-1 upward [downward (p-1)/2 or (p+1)/2].

Original entry on oeis.org

3, 5, 11, 23, 37, 83, 157, 179, 359, 661, 719, 877, 997, 1019, 1237, 1439, 1657, 2039, 2063, 2137, 2459, 2557, 2819, 2903, 2963, 3023, 3061, 3623, 3779, 3803, 3863, 4177, 4261, 4357, 4621, 4919, 5399, 5581, 5639, 6037, 6121, 6217, 6361, 6899, 6983, 7079

Offset: 1

Alexandre Wajnberg, Aug 31 2005

These primes may be part of Cunningham chains longer than three terms. It seems the two operators are never mixed, except for 3, 5 and 7: -for 3, we have: 2 through <2p-1> -> 3 through <2p+1> -> 7 -for 5: 3 <2p-1> -> 5 <2p+1> -> 11 -for 7: 3 <2p+1> -> 7 <2p-1> -> 13
For p > 7, such a mixed chain with p in the middle is impossible because the number 3 would be a nontrivial factor of either the smallest or the largest term. - Jeppe Stig Nielsen, May 05 2019
Primes (excluding 2 and 7) that divide more than one semiprime triangular number A068443. - Jeppe Stig Nielsen, May 05 2019
The disjoint union of A059455 and A109835. - Jeppe Stig Nielsen, May 05 2019

			a(3)=11 is here because 5->11->23 through <2p+1>;
a(4)=23 because 11->23->47 through <2p+1>;
a(5)=37 because 19->37->73 through <2p-1>.

Chris Caldwell's Prime Glossary, Cunningham chains.

Cf. A005382, A005383, A005384, A005385, A059455, A068497.

forprime(p=3,10^6,if(p%3==2,isprime((p-1)/2)&&isprime(2*p+1),isprime((p+1)/2)&&isprime(2*p-1))&&print1(p,", ")) \\ Jeppe Stig Nielsen, May 05 2019

A269668 Smallest k >= 0 such that neither (k + 1)n - k nor (k + 1)n + k is prime.

Original entry on oeis.org

0, 2, 3, 0, 5, 0, 7, 0, 0, 0, 7, 0, 1, 0, 0, 0, 1, 0, 4, 0, 0, 0, 4, 0, 0, 0, 0, 0, 10, 0, 2, 0, 0, 0, 0, 0, 6, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 3

Offset: 1

Juri-Stepan Gerasimov, Mar 03 2016

nonn

Numbers n such that a(n) = m: 2, 31, 41, 131, 157, ... (for m = 2);

3, 53, 83, 97, 139, ... (for m = 3); 19, 23, 79, 191, ... (for m = 4), ...

			For n = 2, k = 0: (0 + 1)*2 - 0 = 2 is prime and (0 + 1)*2 + 0 = 2 is prime; for n = 2, k = 1: (1 + 1)*2 - 1 = 3 is prime and (1 + 1)*2 + 1 = 5 is prime; for n = 2, k = 2: (2 + 1)*2 - 2 = 4 is composite and (2 + 1)*2 + 2 = 6 is composite, so a(2) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A018252 (a(n) = 0), A068497 (a(n) = 1).

Table[SelectFirst[Range[0, 120], And[! PrimeQ[n (# + 1) - #], ! PrimeQ[n (# + 1) + #]] &], {n, 120}] (* Michael De Vlieger, Mar 04 2016, Version 10 *)

A269668(n) = {my(k=0); while (isprime((k+1)*n-k) || isprime((k+1)*n+k), k++); k; } \\ Michel Marcus, Apr 04 2016, corrected by Antti Karttunen, Dec 27 2018

Definition corrected by Michael De Vlieger, Mar 04 2016

A172056 Primes p such that 2p+-1 and 2p+-3 are all composites.

Original entry on oeis.org

59, 61, 103, 109, 149, 151, 163, 257, 313, 389, 401, 449, 479, 541, 569, 571, 673, 677, 709, 733, 769, 821, 823, 839, 857, 883, 919, 947, 971, 983, 1061, 1087, 1093, 1097, 1129, 1151, 1163, 1181, 1249, 1283, 1301, 1319, 1321, 1381, 1433, 1489, 1493, 1549

Offset: 1

Juri-Stepan Gerasimov, Jan 24 2010

nonn

			a(1)=59 because 2*59-1=117, 2*59+1=119, 2*59-3=115, 2*59+3=121 are all composites.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A002808, A068497.

okQ[n_]:=Union[PrimeQ[{2n+1,2n-1,2n+3,2n-3}]]=={False}; Select[Prime[Range[250]],okQ] (* Harvey P. Dale, Feb 07 2010 *)
Select[Prime[Range[300]],AllTrue[2#+{1,3,-1,-3},CompositeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 20 2015 *)

Corrected and extended by Harvey P. Dale, Feb 07 2010

cp's OEIS Frontend

A109998 Non-Cunningham primes: primes isolated from any Cunningham chain under any iteration of 2p+-1 or (p+-1)/2.

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A195685 Primes p for which tau(2p-1) = tau(2p+1) = 4.

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A109927 First primes p connected to two primes either by 2p+1 or 2p-1 upward [downward (p-1)/2 or (p+1)/2].

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A269668 Smallest k >= 0 such that neither (k + 1)*n - k nor (k + 1)*n + k is prime.

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Mathematica

PARI

Extensions

A172056 Primes p such that 2*p+-1 and 2*p+-3 are all composites.

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Mathematica

Extensions

A269668 Smallest k >= 0 such that neither (k + 1)n - k nor (k + 1)n + k is prime.

A172056 Primes p such that 2p+-1 and 2p+-3 are all composites.