A088878 -id:A088878 - cp's OEIS frontend

A091179 A088878 indexed by A000040.

2, 3, 4, 5, 6, 9, 12, 14, 15, 16, 18, 19, 20, 27, 30, 31, 33, 38, 39, 42, 43, 44, 47, 54, 55, 56, 58, 59, 62, 63, 65, 67, 68, 69, 74, 79, 83, 84, 86, 89, 91, 93, 94, 99, 100, 102, 107, 108, 110, 122, 123, 129, 133, 134, 135, 139, 143, 147, 153, 154, 155, 156, 162, 167

Offset: 1

Ray Chandler, Dec 27 2003

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A088878, A091180, A091181.

select(n -> isprime(3*ithprime(n)-2), [$1..1000]); # Robert Israel, Mar 04 2016

PrimePi@ Select[Prime@ Range@ 167, PrimeQ[3 # - 2] &] (* Michael De Vlieger, Mar 04 2016 *)

a(n)=k such that A000040(k) = A088878(n).

Offset corrected by Michael De Vlieger, Mar 04 2016

A265759 Numerators of primes-only best approximates (POBAs) to 1; see Comments.

Original entry on oeis.org

3, 2, 5, 13, 11, 19, 17, 31, 29, 43, 41, 61, 59, 73, 71, 103, 101, 109, 107, 139, 137, 151, 149, 181, 179, 193, 191, 199, 197, 229, 227, 241, 239, 271, 269, 283, 281, 313, 311, 349, 347, 421, 419, 433, 431, 463, 461, 523, 521, 571, 569, 601, 599, 619, 617

Offset: 1

Clark Kimberling, Dec 15 2015

Suppose that x > 0. A fraction p/q of primes is a primes-only best approximate (POBA), and we write "p/q in B(x)", if 0 < |x - p/q| < |x - u/v| for all primes u and v such that v < q. Note that for some choices of x, there are values of q for which there are two POBAs. In these cases, the greater is placed first; e.g., B(3) = (7/2, 5/2, 17/5, 13/5, 23/7, 19/7, ...).
See A265772 and A265774 for definitions of lower POBA and upper POBA. In the following guide, for example, A001359/A006512 represents (conjecturally in some cases) the Lower POBAs p(n)/q(n) to 1, where p = A001359 and q = A006512 except for first terms in some cases. Every POBA is either a lower POBA or an upper POBA.
x        Lower POBA       Upper POBA       POBA
1        A001359/A006512  A006512/A001359  A265759/A265760
3/2      A104163/A158708  A162336/A158709  A265761/A222565
2        A005383/A005382  A005385/A005384  A079149/A120628
3        A091180/A088878  A094525/A023208  A265763/A265764
4        A162857/A062737  A090866/A023212  A265765/A120639
5        A265766/A158318  A265767/A023217  A265768/A265769
6        A227756/A158015  A051644/A007693  A265770/A265771
sqrt(2)  A265772/A265773  A265774/A265775  A265776/A265777
sqrt(3)  A265778/A265779  A265780/A265781  A265782/A265783
sqrt(5)  A265784/A265785  A265786/A265787  A265788/A265789
sqrt(8)  A265790/A265791  A265792/A265793  A265794/A265795
tau      A265796/A265797  A265798/A265799  A265800/A265801
1/tau    A265799/A265798  A265797/A265796  A265806/A265807
pi       A265808/A265809  A265810/A265811  A265812/A265813
e        A265814/A265815  A265816/A265817  A265818/A265819

			The POBAs for 1 start with 3/2, 2/3, 5/7, 13/11, 11/13, 19/17, 17/19, 31/29, 29/31, 43/41, 41/43, 61/59, 59/61. For example, if p and q are primes and q > 13, then 11/13 is closer to 1 than p/q is.

Cf. A000040, A001359, A006512, A265759, A265760.

x = 1; z = 200; p[k_] := p[k] = Prime[k];
t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265759/A265760 *)
Numerator[tL]   (* A001359 *)
Denominator[tL] (* A006512 *)
Numerator[tU]   (* A006512 *)
Denominator[tU] (* A001359 *)
Numerator[y]    (* A265759 *)
Denominator[y]  (* A265760 *)

A091180 Primes of the form 3*p - 2 such that p is a prime.

Original entry on oeis.org

7, 13, 19, 31, 37, 67, 109, 127, 139, 157, 181, 199, 211, 307, 337, 379, 409, 487, 499, 541, 571, 577, 631, 751, 769, 787, 811, 829, 877, 919, 937, 991, 1009, 1039, 1117, 1201, 1291, 1297, 1327, 1381, 1399, 1459, 1471, 1567, 1621, 1669, 1759, 1777, 1801

Offset: 1

Ray Chandler, Dec 27 2003

Mother primes of order 1. - Artur Jasinski, Dec 12 2007

			From _K. D. Bajpai_, Jun 20 2015: (Start)
a(4) = 31: 3*11 - 2 = 31; A088878(4) = 11.
a(6) = 67: 3*23 - 2 = 67; A088878(6) = 23.
(End)

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A088878, A091179, A091181.

Cf. A136020.

[ k: p in PrimesUpTo(1000) | IsPrime(k)  where k is (3*p-2) ]; // K. D. Bajpai, Jun 20 2015

A091180:= n-> (3*ithprime(n)-2): select(isprime,[seq((A091180(n), n=1..100))]);  # K. D. Bajpai, Jun 20 2015

n = 1; a = {}; Do[If[PrimeQ[(Prime[k] + 2n)/(2n + 1)], AppendTo[a, Prime[k]]], {k, 1, 500}]; a (* Artur Jasinski, Dec 12 2007 *)
Select[Table[3*Prime[n] - 2,{n, 1000}], PrimeQ] (* K. D. Bajpai, Jun 20 2015 *)

forprime(p =  1, 1000, k =( 3*p -2); if ( isprime(k), print1(k, ", "))); \\  K. D. Bajpai, Jun 20 2015

a(n) = 3*A088878(n)-2.

Name clarified by Jinyuan Wang, Aug 06 2021

A063908 Numbers k such that k and 2*k-3 are primes.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 23, 31, 37, 41, 43, 53, 67, 71, 83, 97, 101, 107, 113, 127, 137, 157, 167, 181, 191, 193, 211, 223, 233, 241, 251, 263, 283, 311, 317, 331, 347, 373, 421, 431, 433, 443, 457, 461, 487, 521, 547, 563, 577, 587, 613, 617, 631, 641, 643, 647

Offset: 1

N. J. A. Sloane, Aug 31 2001

nonn

If p is in this sequence then the products of positive powers of 3, p and 2p-3 are entries in A086486. - Victoria A Sapko (vsapko(AT)canes.gsw.edu), Sep 23 2003
Median prime of AP3's starting at 3, i.e., triples of primes (3,p,q) in arithmetic progression. - M. F. Hasler, Sep 24 2009
a(n) = sum of the coprimes(p) mod (p+1). - J. M. Bergot, Nov 13 2014
A010051(2*a(n)-3) = 1. - Reinhard Zumkeller, Jul 02 2015
A098090 INTERSECT A000040. - R. J. Mathar, Mar 23 2017

			From _K. D. Bajpai_, Nov 29 2019: (Start)
a(5) = 13 is prime and 2*13 - 3 = 23 is also prime.
a(6) = 17 is prime and 2*17 - 3 = 31 is also prime.
(End)

Harry J. Smith and K. D. Bajpai, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)
Carlos Rivera, Puzzle 34.- Prime Triplets in arithmetic progression, The Prime Puzzles & Problems Connection. [From _M. F. Hasler_, Sep 24 2009]

Cf. A005382.
Cf. A000040, A010051, A088878, A172287.
Cf. A259730.

a063908 n = a063908_list !! (n-1)
a063908_list = filter
   ((== 1) . a010051' . (subtract 3) . (* 2)) a000040_list
-- Reinhard Zumkeller, Jul 02 2015

[n : n in [0..700] | IsPrime(n) and IsPrime(2*n-3)]; // Vincenzo Librandi, Nov 14 2014

select(k -> andmap(isprime, [k, 2*k-3]), [seq(k, k=1.. 10^4)]); # K. D. Bajpai, Nov 29 2019

Select[Prime[Range[6! ]],PrimeQ[2*#-3]&] (* Vladimir Joseph Stephan Orlovsky, Nov 17 2009 *)

{ n=0; p=1; for (m=1, 10^9, p=nextprime(p+1); if (isprime(2*p - 3), write("b063908.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 02 2009

forprime( p=1,default(primelimit), isprime(2*p-3) && print1(p",")) \\ M. F. Hasler, Sep 24 2009

a(n) = A241817(n)/2. - Wesley Ivan Hurt, Apr 08 2018

A125272 Primes p such that 3p - 2 and 3p + 2 are also primes.

Original entry on oeis.org

3, 5, 7, 13, 23, 37, 43, 103, 127, 163, 167, 257, 293, 313, 337, 433, 523, 757, 797, 887, 953, 1013, 1063, 1153, 1283, 1303, 1307, 1483, 1597, 1657, 1667, 1693, 1723, 1783, 1913, 2003, 2333, 2347, 2557, 2897, 2927, 3067, 3533, 3823, 3943, 4003, 4013, 4093

Offset: 1

Zak Seidov, Nov 26 2006

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

Intersection of A023208 and A088878.

Cf. A125215.

[p: p in PrimesUpTo(70000)| IsPrime(3*p-2)and IsPrime(3*p+2)] // Vincenzo Librandi, Jan 29 2011

lst={}; Do[p=Prime[n]; If[PrimeQ[3*p-2]&&PrimeQ[3*p+2],AppendTo[lst,p]],{n,7!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 18 2009 *)
Select[Prime[Range[600]],AllTrue[3#+{2,-2},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 29 2021 *)

is(n)=isprime(3*n-2)&&isprime(3*n+2)&&isprime(n) \\ Charles R Greathouse IV, Jul 02 2013

a(n) = A125215(n)/3.

A136017 a(n) = 36n^2 - 1.

Original entry on oeis.org

35, 143, 323, 575, 899, 1295, 1763, 2303, 2915, 3599, 4355, 5183, 6083, 7055, 8099, 9215, 10403, 11663, 12995, 14399, 15875, 17423, 19043, 20735, 22499, 24335, 26243, 28223, 30275, 32399, 34595, 36863, 39203, 41615, 44099, 46655, 49283, 51983

Offset: 1

Artur Jasinski, Dec 10 2007

The least common multiple of 6*n+1 and 6*n-1. - Colin Barker, Feb 11 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
X. Gourdon and P. Sebah, Collection of series for Pi
Eric Weisstein's World of Mathematics, Pell Equation
Edward Everett Withford, Pell Equation
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A088878, A023208, A136016, A061037, A016910.

[36*n^2 - 1: n in [1..50]]; // Vincenzo Librandi, Jul 09 2012

Mathematica
```
Table[36n^2 - 1, {n, 1, 100}]
```

a(n)=36*n^2-1 \\ Jaume Oliver Lafont, Oct 20 2009

O.g.f.: x*(-35-38*x+x^2)/(-1+x)^3 = 1-35/(-1+x)-108/(-1+x)^2-72/(-1+x)^3. - R. J. Mathar, Dec 12 2007
a(n) = A061037(12n+10)=(6n-1)*(6n+1). - Paul Curtz, Sep 25 2008
Sum_{k>=1} (-1)^(k+1)/a(k) = (Pi-3)/6. - Jaume Oliver Lafont, Oct 20 2009
E.g.f.: 1 + (36 x^2 + 26 x - 1) exp(x). - Robert Israel, Jun 09 2016
Product_{n >= 1} A016910(n) / a(n) = Pi / 3. - Fred Daniel Kline, Jun 09 2016
Sum_{n>=1} 1/a(n) = 1/2 - sqrt(3)*Pi/12. - Amiram Eldar, Jun 27 2020

A136061 Primes p such that (p+4)/5 is also prime.

Original entry on oeis.org

11, 31, 61, 151, 181, 211, 331, 541, 631, 691, 751, 811, 991, 1051, 1201, 1381, 1531, 1741, 1831, 1861, 2161, 2281, 2311, 2731, 2851, 3001, 3061, 3301, 3361, 3541, 3631, 3691, 3931, 4051, 4111, 4261, 4591, 4831, 4951, 5101, 5431, 5581, 5641, 5851, 6151

Offset: 1

Artur Jasinski, Dec 12 2007

Equivalently: Mother primes of order 2. For smallest mother primes of order n see A136020 (also definition). For mother primes of order 1 see A091180.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A088878, A091180, A136019, A136020, A136062, A136063, A136064, A136065, A136066, A136067, A136068, A136069, A136070.

A136061:=Filtered(Filtered([1..10^6],IsPrime),p->IsPrime((p+4)/5)); # Muniru A Asiru, Oct 10 2017

n = 2; a = {}; Do[If[PrimeQ[(Prime[k] + 2n)/(2n + 1)], AppendTo[a, Prime[k]]], {k, 1, 1500}]; a
Select[Prime[Range[400]], PrimeQ[(# + 4) / 5]&] (* Vincenzo Librandi, Apr 14 2013 *)

{forprime(p=1,1e4/*default(primelimit)*/, p%5-1 & next; isprime(p\5+1) & print1(p","))}  \\ M. F. Hasler, Feb 26 2012

A136066 Mother primes of order 7.

Original entry on oeis.org

31, 61, 151, 181, 241, 271, 331, 421, 541, 601, 631, 691, 991, 1051, 1171, 1231, 1321, 1531, 1621, 1951, 2221, 2251, 2341, 2671, 2851, 2971, 3331, 3391, 3571, 3931, 4021, 4051, 4201, 4231, 4591, 4651, 4951, 5281, 5581, 5821, 6121, 6271, 6301, 6451, 6481

Offset: 1

Artur Jasinski, Dec 12 2007

nonn

For smallest mother primes of order n see A136020 (also definition). For mother primes of order 1 see A091180. For mother primes of order 2 see A136061. For mother primes of order 3 see A136062. For mother primes of order 4 see A136063. For mother primes of order 5 see A136064. For mother primes of order 6 see A136065.

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

Cf. A088878, A091180, A136019, A136020, A136061, A136062, A136063, A136064, A136065, A136067, A136068, A136069, A136070.

n = 7; a = {}; Do[If[PrimeQ[(Prime[k] + 2n)/(2n + 1)], AppendTo[a, Prime[k]]], {k, 1, 1500}]; a

A136051 Primes p such that 5*p-4 is also prime.

Original entry on oeis.org

3, 7, 13, 31, 37, 43, 67, 109, 127, 139, 151, 163, 199, 211, 241, 277, 307, 349, 367, 373, 433, 457, 463, 547, 571, 601, 613, 661, 673, 709, 727, 739, 787, 811, 823, 853, 919, 967, 991, 1021, 1087, 1117, 1129, 1171, 1231, 1291, 1297, 1399, 1471, 1483, 1549

Offset: 1

Artur Jasinski, Dec 12 2007

nonn

Previous name: Daughter primes of order 2.

For daughter primes of order 1 see A088878. For smallest daughter primes of order n see A136019 (also definition).

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

Cf. A088878, A091180, A136019, A136020, A136052, A136053, A136054, A136055, A136056, A136057, A136058, A136059, A136060.

n = 2; a = {}; Do[If[PrimeQ[(Prime[k] + 2n)/(2n + 1)], AppendTo[a, (Prime[k] + 2n)/(2n + 1)]], {k, 1, 1500}]; a
(* Second program: *)
Select[Prime@ Range@ 250, PrimeQ[5 # - 4] &] (* Michael De Vlieger, Aug 04 2017 *)

lista(nn) = forprime(p=2, nn, if (isprime(5*p-4), print1(p, ", ")))

New name from Michel Marcus, Aug 04 2017

A136052 Daughter primes of order 3.

Original entry on oeis.org

5, 7, 11, 17, 19, 29, 31, 41, 61, 67, 71, 79, 89, 97, 101, 107, 109, 127, 131, 137, 139, 151, 157, 167, 197, 211, 227, 229, 239, 269, 277, 307, 317, 331, 347, 349, 379, 401, 409, 419, 431, 439, 449, 461, 479, 509, 547, 601, 607, 619, 641, 647, 661, 677, 691

Offset: 1

Artur Jasinski, Dec 12 2007

For daughter primes of order 1 see A088878. For daughter primes of order 2 see A136051. For smallest daughter primes of order n see A136019 (also definition)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A088878, A091180, A136019, A136020, A136052, A136053, A136054, A136055, A136056, A136057, A136058, A136059, A136060.

n = 3; a = {}; Do[If[PrimeQ[(Prime[k] + 2n)/(2n + 1)], AppendTo[a, (Prime[k] + 2n)/(2n + 1)]], {k, 1, 1500}]; a

cp's OEIS Frontend

A091179 A088878 indexed by A000040.

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A265759 Numerators of primes-only best approximates (POBAs) to 1; see Comments.

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A091180 Primes of the form 3*p - 2 such that p is a prime.

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A063908 Numbers k such that k and 2*k-3 are primes.

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A125272 Primes p such that 3p - 2 and 3p + 2 are also primes.

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Magma

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A136017 a(n) = 36n^2 - 1.

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A136061 Primes p such that (p+4)/5 is also prime.

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