A136020 -id:A136020 - cp's OEIS frontend

A136019 Smallest prime of the form (prime(k)+2n)/(2n+1), any k.

3, 3, 5, 3, 3, 5, 3, 7, 11, 3, 3, 5, 5, 3, 11, 3, 3, 5, 3, 3, 5, 5, 7, 5, 3, 3, 7, 5, 13, 7, 3, 3, 5, 3, 13, 5, 3, 7, 5, 3, 3, 13, 5, 3, 7, 5, 3, 5, 3, 7, 7, 3, 7, 11, 3, 3, 5, 11, 3, 7, 7, 3, 5, 11, 3, 13, 3, 7, 5, 3, 7, 11, 7, 13, 7, 3, 3, 11, 23, 7, 5, 3, 31, 5, 13, 3, 5, 5, 3, 7, 3, 13, 7, 3, 3, 5, 7

Offset: 1

Artur Jasinski, Dec 10 2007

The associated prime(k) are in A136020.

			a(1)=3 because 3 is smallest prime of the form (p+2)/3; in this case prime(k)=7.
a(2)=3 because 3 is smallest prime of the form (p+4)/5; in this case prime(k)=11.
a(3)=5 because 5 is smallest prime of the form (p+6)/7; in this case prime(k)=29.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A136020, A136026, A136027.

N:= 10^5: # to allow prime(k) <= N
Primes:= select(isprime,[2,seq(2*i+1,i=1..floor((N-1)/2))]):
f:= proc(t,n)
  local s;
  s:= (t+2*n)/(1+2*n);
  type(s,integer) and isprime(s)
end proc:
for n from 1 do
  p:= ListTools:-SelectFirst(f, Primes,n);
  if p = NULL then break fi;
  A[n]:= (p+2*n)/(1+2*n);
od:
seq(A[i],i=1..n-1); # Robert Israel, Sep 08 2014

a = {}; Do[k = 1; While[ !PrimeQ[(Prime[k] + 2n)/(2n + 1)], k++ ]; AppendTo[a, (Prime[k] + 2n)/(2n + 1)], {n, 1, 200}]; a
sp[n_]:=Module[{k=1},While[!PrimeQ[(Prime[k]+2n)/(2n+1)],k++];(Prime[ k]+2n)/(2n+1)]; Array[sp,100] (* Harvey P. Dale, May 20 2021 *)

a(n)=my(N=2*n,k=0,t);forprime(p=2,default(primelimit),k++;t=(p+N)/(N+1);if(denominator(t)==1&isprime(t),return(t))) \\ Charles R Greathouse IV, Jun 16 2011

Edited by R. J. Mathar, May 17 2009

A088878 Prime numbers p such that 3p - 2 is a prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 23, 37, 43, 47, 53, 61, 67, 71, 103, 113, 127, 137, 163, 167, 181, 191, 193, 211, 251, 257, 263, 271, 277, 293, 307, 313, 331, 337, 347, 373, 401, 431, 433, 443, 461, 467, 487, 491, 523, 541, 557, 587, 593, 601, 673, 677, 727, 751, 757, 761

Offset: 1

Giovanni Teofilatto, Nov 27 2003

Indices of semiprime octagonal numbers. - Jonathan Vos Post, Feb 16 2006
Daughter primes of order 1. - Artur Jasinski, Dec 12 2007
A010051(3*a(n)-2) = 1. - Reinhard Zumkeller, Jul 02 2015

			For p = 3, 3p - 2 = 7;
for p = 523, 3p - 2 = 1567.

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta, UTET, CittaStudiEdizioni, Milano 1997

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Octagonal Number.
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A000040, A000567, A001222, A001358, A091179, A091180, A091181, A136019, A136020, A153183, A153184.
Cf. A010051, A063908, A172287.
Cf. A259730.

a088878 n = a088878_list !! (n-1)
a088878_list = filter ((== 1) . a010051' . subtract 2 . (* 3)) a000040_list
-- Reinhard Zumkeller, Jul 02 2015

[ p: p in PrimesUpTo(770) | IsPrime(3*p-2) ]; // Klaus Brockhaus, Dec 21 2008

lst={};Do[p=Prime[n];If[PrimeQ[3*p-2],AppendTo[lst,p]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 22 2008 *)
n = 1; a = {}; Do[If[PrimeQ[(Prime[k] + 2n)/(2n + 1)], AppendTo[a, (Prime[k] + 2n)/(2n + 1)]], {k, 1, 500}]; a (* Artur Jasinski, Dec 12 2007 *)
Select[Prime[Range[150]],PrimeQ[3#-2]&] (* Harvey P. Dale, Feb 27 2024 *)

list(lim)=select(p->isprime(3*p-2),primes(primepi(lim))) \\ Charles R Greathouse IV, Jul 25 2011

Corrected and extended by Ray Chandler, Dec 27 2003

Entry revised by N. J. A. Sloane, Nov 28 2006, Jul 08 2010

A023208 Primes p such that 3*p + 2 is also prime.

Original entry on oeis.org

3, 5, 7, 13, 17, 19, 23, 29, 37, 43, 59, 79, 83, 89, 97, 103, 127, 139, 149, 163, 167, 173, 197, 199, 227, 233, 239, 257, 269, 293, 313, 317, 337, 349, 353, 367, 383, 397, 409, 419, 433, 439, 457, 479, 499, 503, 523, 569, 577, 607, 643, 659, 709, 757, 769, 797, 859, 863

Offset: 1

David W. Wilson

Also, son primes of order 1. For smallest son primes of order n see A136027 (also definition). For son primes of order 2 see A136082. - Artur Jasinski, Dec 12 2007

Zak Seidov and Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 1000 terms from _Zak Seidov_)
Rosemary Sullivan and Neil Watling, Independent divisibility pairs on the set of integers from 1 to n, INTEGERS 13 (2013) #A65.

Cf. A023208, A094524, A136019, A136020, A136026, A136027, A136082, A136083, A136084, A136085, A136086, A136087, A136088, A136089, A136090, A136091.

a023208 n = a023208_list !! (n-1)
a023208_list = filter ((== 1) . a010051 . (+ 2) . (* 3)) a000040_list
-- Reinhard Zumkeller, Aug 15 2011

[n: n in PrimesUpTo(900) | IsPrime(3*n+2)]; // Vincenzo Librandi, Nov 20 2010

n = 1; a = {}; Do[If[PrimeQ[(Prime[k] - 2n)/(2n + 1)], AppendTo[a, (Prime[k] - 2n)/(2n + 1)]], {k, 1, 1000}]; a (* Artur Jasinski, Dec 12 2007 *)

isA023208(n) = isprime(n) && isprime(3*n+2) \\ Michael B. Porter, Jan 30 2010

Edited by N. J. A. Sloane, May 16 2008 at the suggestion of R. J. Mathar

A136026 Smallest prime of the form (2n+1)p + 2n with p prime.

Original entry on oeis.org

11, 19, 41, 53, 43, 103, 59, 67, 113, 83, 137, 149, 107, 173, 433, 131, 139, 443, 233, 163, 257, 179, 281, 293, 1019, 211, 439, 227, 353, 487, 251, 389, 401, 827, 283, 1021, 449, 307, 631, 647, 331, 509, 347, 1601, 727, 557, 379, 1163, 593, 2423, 617, 419, 641

Offset: 1

Artur Jasinski, Dec 10 2007

The associated p are in A136027.

			a(1)=11 because 11 is smallest prime p such that (p-2)/3 is prime.
a(2)=19 because 19 is smallest prime p such that (p-4)/5 is prime.
a(3)=41 because 41 is smallest prime p such that (p-6)/7 is prime.

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

Cf. A136019, A136020, A136027.

a = {}; Do[k = 1; While[ !PrimeQ[(Prime[k] - 2n)/(2n + 1)], k++ ]; AppendTo[a, Prime[k]], {n, 1, 100}]; a

a(n)=my(t);forprime(p=2,,if(isprime(t=2*n*(p+1)+p),return(t))) \\ Charles R Greathouse IV, Mar 21 2013

Edited by R. J. Mathar, May 17 2009

A136027 Smallest prime of the form (p-2n)/(2n+1) with p prime.

Original entry on oeis.org

3, 3, 5, 5, 3, 7, 3, 3, 5, 3, 5, 5, 3, 5, 13, 3, 3, 11, 5, 3, 5, 3, 5, 5, 19, 3, 7, 3, 5, 7, 3, 5, 5, 11, 3, 13, 5, 3, 7, 7, 3, 5, 3, 17, 7, 5, 3, 11, 5, 23, 5, 3, 5, 5, 3, 5, 7, 3, 11, 7, 3, 3, 5, 5, 3, 5, 5, 3, 11, 3, 3, 13, 3, 11, 11, 7, 3, 5, 5, 3, 5, 3, 11, 5, 3, 3, 5, 5

Offset: 1

Artur Jasinski, Dec 10 2007

The associated p are in A136026.

			a(1)=3 because 3 is smallest prime of the form (p-2)/3; in this case prime(k)=11.
a(2)=3 because 3 is smallest prime of the form (p-4)/5; in this case prime(k)=19.
a(3)=5 because 5 is smallest prime of the form (p-6)/7; in this case prime(k)=41.

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

Cf. A136019, A136020, A136026.

a = {}; Do[k = 1; While[ !PrimeQ[(Prime[k] - 2n)/(2n + 1)], k++ ]; AppendTo[a, (Prime[k] - 2n)/(2n + 1)], {n, 1, 100}]; a

a(n)=forprime(p=2,,if(isprime(2*n*(p+1)+p),return(p))) \\ Charles R Greathouse IV, Mar 21 2013

Edited by R. J. Mathar, May 17 2009

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

A139075 Primes p arising in A139074.

Original entry on oeis.org

3, 2, 3, 31, 1009, 2, 5702401, 631

Offset: 1

Artur Jasinski, Apr 08 2008, Apr 21 2008

nonn

a(23) = (23+1579!)/23. - Andrew V. Sutherland, Apr 11 2008.
Smallest mother factorial prime p of order n, i.e. smallest prime of the form (p!+n)/n where p is prime.
For smallest daughter factorial prime p of order n see A139074.
For smallest father factorial prime p of order n see A139207.
For smallest son factorial prime p of order n see A139206.
a(9)=26737!/9+1 is a 106758 digit (probable) prime.  Easily calculated but too large to enter here a(10)=13, a(11)=566092801, a(12)=11. [Robert Price, Jan 19 2011]

Cf. A082672, A089085, A089130, A117141, A007749, A139056-A139066, A020458, A139068, A137390, A139070-A139075, A136019, A136020, A136026, A136027.

a = {}; Do[k = 1; While[ ! PrimeQ[(Prime[k]! + n)/n], k++ ]; AppendTo[a, Prime[(Prime[k]! + n)/n]], {n, 1, 8}]; a

A136082 Son primes of order 5.

Original entry on oeis.org

3, 11, 17, 23, 41, 53, 59, 107, 131, 167, 173, 179, 191, 257, 263, 269, 389, 401, 431, 461, 467, 479, 521, 563, 569, 599, 647, 653, 677, 683, 719, 773, 821, 839, 857, 887, 947, 971, 1031, 1049, 1061, 1091, 1103, 1151, 1181, 1217, 1223, 1259, 1277, 1301

Offset: 1

Artur Jasinski, Dec 12 2007

nonn

For smallest son primes of order n see A136027 (also definition). For son primes of order 1 see A023208. For son primes of order 2 see A023218. For son primes of order 3 see A023225. For son primes of order 4 see A023235.

Numbers in this sequence are those primes p such that 11*p + 10 is also prime. Generally, son primes of order n are the primes p such that (2n+1)*p + 2n is also prime. - Bob Selcoe, Apr 04 2015

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

Cf. A023208, A023218, A023225, A023235, A094524, A136019, A136020, A136026, A136027, A023208, A136083, A136084, A136085, A136086, A136087, A136088, A136089, A136090, A136091.

n = 5; a = {}; Do[If[PrimeQ[(Prime[k] - 2n)/(2n + 1)], AppendTo[a, (Prime[k] - 2n)/(2n + 1)]], {k, 1, 1000}]; a
q=10;lst={};Do[p=Prime[n];If[PrimeQ[(q+1)*p+q],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 10 2009 *)
Select[Prime[Range[250]],PrimeQ[11#+10]&] (* Harvey P. Dale, Aug 07 2021 *)

A136083 Son primes of order 6.

Original entry on oeis.org

7, 13, 17, 23, 29, 43, 53, 67, 79, 83, 109, 113, 127, 149, 157, 163, 179, 193, 227, 233, 239, 277, 283, 293, 307, 317, 347, 349, 359, 367, 373, 433, 449, 457, 487, 503, 557, 563, 587, 619, 647, 653, 673, 727, 739, 769, 773, 787, 809, 823, 829, 883, 919, 947

Offset: 1

Artur Jasinski, Dec 12 2007

nonn

For smallest son primes of order n see A136027 (also definition). For son primes of order 1 see A023208. For son primes of order 2 see A023218. For son primes of order 3 see A023225. For son primes of order 4 see A023235. For son primes of order 5 see A136082.

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

Cf. A023208, A023218, A023225, A023235, A094524, A136019, A136020, A136026, A136027, A023208, A136082, A136084, A136085, A136086, A136087, A136088, A136089, A136090, A136091.

n = 6; a = {}; Do[If[PrimeQ[(Prime[k] - 2n)/(2n + 1)], AppendTo[a, (Prime[k] - 2n)/(2n + 1)]], {k, 1, 1000}]; a
q=12;lst={};Do[p=Prime[n];If[PrimeQ[(q+1)*p+q],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 10 2009 *)

A139089 a(n) = prime(n)!/9 + 1.

Original entry on oeis.org

561, 4435201, 691891201, 39520825344001, 13516122267648001, 2872446304320552960001, 982417999304411328282624000001, 913648739353102535302840320000001

Offset: 4

Artur Jasinski, Apr 08 2008

nonn

Cf. A082672, A089085, A089130, A117141, A007749, A139056-A139066, A020458, A139068, A137390, A139070-A139075, A136019, A136020, A136026, A136027, A139089-A139092.

Table[(Prime[n]! + 9)/9, {n, 4, 30}]
Prime[Range[4,12]]!/9+1 (* Harvey P. Dale, Aug 22 2020 *)

cp's OEIS Frontend

A136019 Smallest prime of the form (prime(k)+2*n)/(2*n+1), any k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A088878 Prime numbers p such that 3p - 2 is a prime.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Extensions

A023208 Primes p such that 3*p + 2 is also prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Extensions

A136026 Smallest prime of the form (2n+1)p + 2n with p prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A136027 Smallest prime of the form (p-2n)/(2n+1) with p prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A136019 Smallest prime of the form (prime(k)+2n)/(2n+1), any k.