A070155 -id:A070155 - cp's OEIS frontend

A242720 Smallest even k such that the pair {k-3,k-1} is not a twin prime pair and lpf(k-1) > lpf(k-3) >= prime(n), where lpf = least prime factor (A020639).

12, 38, 80, 212, 224, 440, 440, 854, 1250, 1460, 1742, 2282, 2282, 3434, 4190, 4664, 4760, 4760, 6890, 8054, 8054, 8054, 12374, 12830, 12830, 13592, 13592, 14282, 17402, 17402, 18212, 22502, 22502, 22502, 25220, 28202, 28202, 32234, 32402, 32402, 38012

Offset: 2

Vladimir Shevelev, May 21 2014

nonn

The sequence is nondecreasing. See comment in A242758.
a(n) >= prime(n)^2+3. Conjecture: a(n) <= prime(n)^4. - Vladimir Shevelev, Jun 01 2014
Conjecture. There are only a finite number of composite numbers of the form a(n)-1. Peter J. C. Moses found only two: a(16)-1 = 4189 = 59*71 and a(20)-1 = 6889 = 83^2 and no others up to a(2501). Most likely, there are no others. - Vladimir Shevelev, Jun 09 2014

Peter J. C. Moses, Table of n, a(n) for n = 2..2501
V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2009-2014, (Section 10).

Cf. A020639, A001359, A006512, A070155, A242489, A242490, A242758, A242847, A246748, A246821, A246824.

lpf[n_] := FactorInteger[n][[1, 1]];
Clear[a]; a[n_] := a[n] = For[k = If[n <= 2, 2, a[n-1]], True, k = k+2, If[Not[PrimeQ[k-3] && PrimeQ[k-1]] && lpf[k-1] > lpf[k-3] >= Prime[n], Return[k]]];
Table[a[n], {n, 2, 50}] (* Jean-François Alcover, Nov 02 2018 *)

lpf(k) = factorint(k)[1,1];
vector(60, n, k=6; while((isprime(k-3) && isprime(k-1)) || lpf(k-1)<=lpf(k-3) || lpf(k-3)Colin Barker, Jun 01 2014

Conjecturally, a(n) ~ (prime(n))^2, as n goes to infinity (cf. A246748, A246821). - Vladimir Shevelev, Sep 02 2014

For n>=3, a(n) >= (prime(n)+1)^2 + 2. Equality holds for terms of A246824. - Vladimir Shevelev, Sep 04 2014

A157468 Primes of the form sqrt(p-1)-1, where p is a prime.

Original entry on oeis.org

3, 5, 13, 19, 23, 53, 73, 83, 89, 109, 149, 179, 223, 229, 239, 263, 269, 283, 313, 349, 383, 419, 439, 443, 463, 569, 593, 643, 653, 673, 739, 859, 863, 919, 929, 1009, 1069, 1093, 1123, 1289, 1319, 1373, 1409, 1429, 1433, 1439, 1459

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 01 2009

nonn

			3 is in the sequence because 3 = sqrt(17 - 1) - 1, where 17 is prime.
5 is in the sequence because 5 = sqrt(37 - 1) - 1, where 37 is prime.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A070155, A127435, A127436, A157467.

Column k=1 of A238048 and A238086.

Select[Sqrt[#-1]-1&/@Prime[Range[200000]],PrimeQ]  (* Harvey P. Dale, May 19 2012 *)

A070156 Numbers k such that k-1, k+1, k^2+1 and k^4+1 are all prime numbers.

Original entry on oeis.org

4, 6, 180, 3390, 18060, 19380, 34500, 92220, 92640, 96180, 114660, 127680, 133980, 140760, 159630, 161880, 172170, 207480, 254280, 255840, 263820, 296910, 309780, 378570, 380880, 397590, 408690, 422880, 432660, 440550, 511170, 540390, 572940

Offset: 1

Labos Elemer, Apr 23 2002

			For n = 6: 5, 7, 37, 1297 are all primes.

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

Subsequence of A070155.

Cf. A001359.

Do[s=n; If[PrimeQ[n-1]&&PrimeQ[n+1]&&PrimeQ[1+n^2] &&PrimeQ[1+n^4], Print[n]], {n, 1, 1000000}]

is(k) = isprime(k-1) && isprime(k+1) && isprime(k^2+1) && isprime(k^4+1); \\ Amiram Eldar, Jun 21 2024

Offset corrected by Amiram Eldar, Jun 21 2024

A070025 At these values of k, the 1st, 2nd, 3rd and 4th cyclotomic polynomials all give prime numbers.

Original entry on oeis.org

6, 150, 2730, 9000, 9240, 35280, 41760, 43050, 53280, 65520, 76650, 96180, 111030, 148200, 197370, 207480, 213360, 226380, 254280, 264600, 309480, 332160, 342450, 352740, 375450, 381990, 440550, 458790, 501030, 527070, 552030, 642360, 660810

Offset: 1

Labos Elemer, May 07 2002

Numbers k such that k-1, k+1, k^2+k+1 and k^2+1 are all primes.

			For k = 6: 5, 7, 43 and 37 are prime values of the first 4 cyclotomic polynomials.

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

Cf. A070155, A070156, A070157, A000068, A006313, A006314, A006315, A006316, A056993, A056994, A056995, A005574, A057465, A057002, A070020, A070042.

lst={};Do[If[PrimeQ[n-1]&&PrimeQ[n+1]&&PrimeQ[1+n+n^2]&&PrimeQ[1+n^2], AppendTo[lst, n]], {n, 10^6}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 19 2008 *)
Select[Range[10^6], Function[k, AllTrue[Cyclotomic[#, k] & /@ Range@ 4, PrimeQ]]] (* Michael De Vlieger, Jul 18 2017 *)

is(k) = isprime(k-1) && isprime(k+1) && isprime(k^2+1) && isprime(k^2+k+1); \\ Amiram Eldar, Sep 24 2024

A129293 Numbers m such that m^4-1 has no divisors d with 1 < d < m-1.

Original entry on oeis.org

3, 4, 6, 150, 180, 240, 270, 420, 570, 1290, 1320, 2310, 2550, 2730, 3360, 3390, 4260, 4650, 5850, 5880, 6360, 6780, 9000, 9240, 9630, 10530, 10890, 11970, 13680, 13830, 14010, 14550, 16230, 16650, 18060, 18120, 18540, 19140, 19380, 21600, 21840, 23370

Offset: 1

Reinhard Zumkeller, Apr 09 2007

nonn

Essentially the same as A070155, since m^4-1=(m-1)(m+1)(1+m^2). - R. J. Mathar, Jun 14 2008

			{1,5,7,35,37,185,259,1295} is the set of divisors of 6^4-1, therefore 6 is a term, A129292(6) = #{1,3} = 2.

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

Cf. A070155, A129292, A129295, A129297.

is(k) = k == 3 || (isprime(k-1) && isprime(k+1) && isprime(k^2+1)); \\ Amiram Eldar, Apr 15 2024

A129292(a(n)) = #{1, a(n)-1} = 2.

A070020 At these values of k the first, 2nd and 3rd cyclotomic polynomials all give prime numbers.

Original entry on oeis.org

6, 12, 138, 150, 192, 348, 642, 1020, 1092, 1230, 1620, 1788, 1932, 2112, 2142, 2238, 2658, 2688, 2730, 3330, 3540, 3918, 4002, 4158, 5010, 5640, 6090, 6450, 6552, 6702, 7950, 8088, 9000, 9042, 9240, 9462, 9768, 10008, 10092, 10272, 10302, 10332

Offset: 1

Labos Elemer, May 07 2002

Numbers k such that k-1, k+1 and k^2+k+1 are all primes.

			For k = 6: 5, 7 and 43 are prime values of the first 3 cyclotomic polynomials.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Michael De Vlieger)

Cf. A070155, A070156, A070157, A000068, A006313, A006314, A006315, A006316, A056993, A056994, A056995, A005574, A057465, A057002, A070025, A070042.

psQ[n_]:=And@@PrimeQ[{n-1,n+1,n^2+n+1}]; Select[Range[11000],psQ] (* Harvey P. Dale, Nov 05 2011 *)
Select[Range[10500], AllTrue[Cyclotomic[Range@ 3, #], PrimeQ] &] (* Michael De Vlieger, Dec 08 2018 *)

is(k) = isprime(k-1) && isprime(k+1) && isprime(k^2+k+1); \\ Amiram Eldar, Sep 24 2024

A070157 Numbers k such that k-1, k+1, k^2+1, k^4+1 and k^8+1 are all prime numbers.

Original entry on oeis.org

4, 19380, 9443670, 11054760, 15992070, 22482330, 32557380, 51102510, 57978840, 60549240, 64671570, 84045960, 89757960, 111316170, 112821690, 116433510, 171124380, 171418650, 183082350, 196694760, 197021160, 241803240, 266498460

Offset: 1

Labos Elemer, Apr 23 2002

nonn

			19380 is a term since 19380-1 = 19379, 19380+1 = 19381, 19380^2+1 = 375584401, 19380^4+1 = 141063641523360001 and 19380^8+1 = 19898950959831015581425689600000001 are primes.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Subsequence of A070155 and A070156.

Do[p = Prime[n] + 1; If[ PrimeQ[p + 1] && PrimeQ[1 + p^2] && PrimeQ[1 + p^4] && PrimeQ[1 + p^8], Print[p]], {n, 1, 115000000}]
Select[Range[2665*10^5],AllTrue[{#-1,#+1,#^2+1,#^4+1,#^8+1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 03 2019 *)

is(k) = isprime(k-1) && isprime(k+1) && isprime(k^2+1) && isprime(k^4+1) && isprime(k^8+1); \\ Amiram Eldar, Jun 26 2024

Edited and extended by Robert G. Wilson v, May 04 2002

A070042 At these values of k the 1st, 2nd, 3rd, 4th and 5th cyclotomic polynomials all give prime numbers.

Original entry on oeis.org

1068630, 1441590, 1867950, 3429300, 4084230, 5651730, 6322890, 6770610, 7158630, 7804830, 9437760, 9624270, 13625850, 23194860, 25848840, 26588520, 28714950, 29451840, 32984430, 33650580, 36500910, 38177130, 42856590, 49531020, 50016540, 50222070, 52083330, 54637590

Offset: 1

Labos Elemer, May 07 2002

Numbers k such that C1(k) = k-1, C2(k) = k+1, C3(k) = k^2+k+1, C4(k) = k^2+1 and C5(k) = k^4+k^3+k^2+k+1 are all primes.

			For k = 1068630: the 1st, 2nd, 3rd, 4th and 5th cyclotomic polynomials give a quintet of primes: {1068629, 1068631, 1141971145531, 1141970076901, 1304096876879617162402531}.

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

Cf. A070155, A070156, A070157, A000068, A006313, A006314, A006315, A006316, A056993, A056994, A056995, A005574, A057465, A057002, A070020, A070025.

is(k) = isprime(k-1) && isprime(k+1) && isprime(k^2+1) && isprime(k^2+k+1) && isprime(k^4+k^3+k^2+k+1) ; \\ Amiram Eldar, Sep 24 2024

More terms from Don Reble, May 11 2002

a(24)-a(28) from Amiram Eldar, Sep 24 2024

cp's OEIS Frontend

A242720 Smallest even k such that the pair {k-3,k-1} is not a twin prime pair and lpf(k-1) > lpf(k-3) >= prime(n), where lpf = least prime factor (A020639).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A157468 Primes of the form sqrt(p-1)-1, where p is a prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A070156 Numbers k such that k-1, k+1, k^2+1 and k^4+1 are all prime numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A070025 At these values of k, the 1st, 2nd, 3rd and 4th cyclotomic polynomials all give prime numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A129293 Numbers m such that m^4-1 has no divisors d with 1 < d < m-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A070020 At these values of k the first, 2nd and 3rd cyclotomic polynomials all give prime numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A070157 Numbers k such that k-1, k+1, k^2+1, k^4+1 and k^8+1 are all prime numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A070042 At these values of k the 1st, 2nd, 3rd, 4th and 5th cyclotomic polynomials all give prime numbers.

Views

Author

Keywords