A049481 -id:A049481 - cp's OEIS frontend

A156104 Primes p such that p+36 is also prime.

5, 7, 11, 17, 23, 31, 37, 43, 47, 53, 61, 67, 71, 73, 101, 103, 113, 127, 131, 137, 157, 163, 191, 193, 197, 227, 233, 241, 257, 271, 277, 281, 311, 313, 317, 331, 337, 347, 353, 373, 383, 397, 421, 431, 443, 463, 467, 487, 521, 541, 557, 563, 571, 577, 607

Offset: 1

Vincenzo Librandi, Feb 08 2009

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

Cf. A156112.

Cf. sequences of the type p+n are primes: A001359 (n=2), A023200 (n=4), A023201 (n=6), A023202 (n=8), A023203 (n=10), A046133 (n=12), A153417 (n=14), A049488 (n=16), A153418 (n=18), A153419 (n=20), A242476 (n=22), A033560 (n=24), A252089 (n=26), A252090 (n=28), A049481 (n=30), A049489 (n=32), A252091 (n=34), this sequence (n=36); A062284 (n=50), A049490 (n=64), A156105 (n=72), A156107 (n=144).

[p: p in PrimesUpTo(1000) | IsPrime(p + 36)]; // Vincenzo Librandi, Oct 31 2012

Select[Prime[Range[1000]], PrimeQ[(#+ 36)]&] (* Vincenzo Librandi, Oct 31 2012 *)

A049482 Primes p such that p + 210 is also prime.

Original entry on oeis.org

13, 17, 19, 23, 29, 31, 41, 47, 53, 59, 61, 67, 71, 73, 83, 97, 101, 103, 107, 127, 137, 139, 149, 157, 163, 173, 179, 191, 199, 211, 223, 229, 233, 239, 251, 257, 269, 277, 281, 293, 311, 313, 331, 337, 347, 353, 359, 367, 383, 389, 397, 409, 421, 431, 433

Offset: 1

Labos Elemer

nonn

			Both 13 and 13 + 210 = 223 are prime.

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

Cf. A045320, A001359, A023201, A049481.

Select[Prime@ Range@ 84, PrimeQ[# + 210] &] (* Michael De Vlieger, Jun 29 2017 *)

list(lim)=my(v=List()); forprime(p=2,lim, if(isprime(p+210), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Feb 23 2017

a(n) >> n log^2 n. - Charles R Greathouse IV, Feb 23 2017

A049485 Primes p such that p + 510510 is also prime, where 510510 is the 7th primorial number A002110(7).

Original entry on oeis.org

19, 41, 43, 59, 71, 73, 79, 101, 103, 107, 109, 167, 173, 181, 197, 199, 241, 257, 263, 283, 293, 307, 313, 317, 337, 379, 397, 409, 421, 431, 433, 479, 491, 503, 509, 523, 547, 577, 599, 601, 613, 641, 643, 653, 659, 661, 683, 691, 701, 727, 733, 751, 769

Offset: 1

Labos Elemer

nonn

p and p+510510 are not necessarily consecutive primes.

			19 is a term since it is prime and 19 + 510510 = 510529 is also prime.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A002110, A045320.

Cf. A001359, A023201, A049481, A049482, A049483, A049484, A154114.

lst={};Do[p=Prime[n];If[PrimeQ[p+510510],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 04 2009 *)

isok(p) = isprime(p) && isprime(p + 510510); \\ Amiram Eldar, Mar 15 2025

A154114 Primes p such that p + 9699690 is also prime, where 9699690 is the 8th primorial number A002110(8).

Original entry on oeis.org

23, 37, 41, 43, 59, 73, 79, 83, 109, 113, 127, 137, 151, 163, 197, 199, 223, 227, 229, 233, 239, 251, 263, 269, 283, 313, 337, 349, 373, 383, 389, 409, 421, 449, 457, 463, 479, 523, 557, 599, 617, 647, 691, 727, 739, 743, 751, 757, 761, 773, 797, 811, 821

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 04 2009

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2000 from G. C. Greubel)

Cf. A001359, A002110, A023201, A049481, A049482, A049483, A049484, A049485.

[p: p in PrimesUpTo(1000) | IsPrime(p+9699690)]; // Vincenzo Librandi, Sep 02 2016

lst={};Do[p=Prime[n];If[PrimeQ[p+9699690],AppendTo[lst,p]],{n,6!}];lst
Select[Prime[Range[200]],PrimeQ[#+9699690]&]  (* Harvey P. Dale, Apr 26 2011 *)

is(n)=isprime(n+9699690) && isprime(n) \\ Charles R Greathouse IV, Sep 02 2016

a(n) >> n log^2 n. - Charles R Greathouse IV, Sep 02 2016

A049483 Primes p such that p + 2310 is also prime, where 2310 is the 5th primorial number A002110(5).

Original entry on oeis.org

23, 29, 31, 37, 41, 47, 61, 67, 71, 73, 79, 83, 89, 101, 107, 113, 127, 131, 137, 149, 157, 163, 167, 193, 211, 229, 233, 239, 241, 269, 281, 283, 307, 311, 337, 347, 349, 353, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 439, 443, 457, 467, 479

Offset: 1

Labos Elemer

nonn

p and p+2310 are not necessarily consecutive primes.

			23 is a term since it is prime and 23 + 2310 = 2333 is also prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A002110, A045320.

Cf. A001359, A023201, A049481, A049482, A049484, A049485, A154114.

Select[Prime[Range[100]],PrimeQ[#+2310]&] (* Harvey P. Dale, Nov 15 2012 *)

isok(p) = isprime(p) && isprime(p + 2310); \\ Amiram Eldar, Mar 15 2025

A049484 Primes p such that p + 30030 is also prime, where 30030 is the 6th primorial number A002110(6).

Original entry on oeis.org

17, 29, 41, 59, 61, 67, 73, 79, 83, 89, 103, 107, 109, 131, 139, 151, 157, 167, 173, 181, 193, 211, 223, 229, 239, 241, 263, 277, 283, 293, 311, 317, 337, 359, 373, 397, 401, 419, 439, 461, 463, 467, 479, 487, 499, 509, 523, 547, 563, 601, 607, 613, 619, 631

Offset: 1

Labos Elemer

nonn

p and p+30030 are not necessarily consecutive primes.

			17 is a term since it is prime and 17 + 30030 = 30047 is also prime.

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

Cf. A002110, A045320.

Cf. A001359, A023201, A049481, A049482, A049483, A049485, A154114.

Select[Prime[Range[150]],PrimeQ[#+30030]&] (* Harvey P. Dale, Sep 21 2022 *)

isok(p) = isprime(p) && isprime(p + 30030); \\ Amiram Eldar, Mar 15 2025

A252089 Primes p such that p + 26 is prime.

Original entry on oeis.org

3, 5, 11, 17, 41, 47, 53, 71, 83, 101, 113, 131, 137, 167, 173, 197, 251, 257, 281, 311, 347, 353, 383, 431, 461, 521, 587, 593, 617, 647, 683, 701, 743, 761, 797, 827, 857, 881, 911, 941, 971, 983, 1013, 1061, 1091, 1097, 1103, 1187, 1223, 1277, 1301, 1373

Offset: 1

Vincenzo Librandi, Dec 14 2014

			17 is in this sequence because 17+26 = 43 is prime.
431 is in this sequence because 431+26 = 457 is prime.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. sequences of the type p+n are primes: A001359 (n=2), A023200 (n=4), A023201 (n=6), A023202 (n=8), A023203 (n=10), A046133 (n=12), A153417 (n=14), A049488 (n=16), A153418 (n=18), A153419 (n=20), A242476 (n=22), A033560 (n=24), this sequence (n=26), A252090 (n=28), A049481 (n=30), A049489 (n=32), A252091 (n=34), A156104 (n=36); A062284 (n=50), A049490 (n=64), A156105 (n=72), A156107 (n=144).

[NthPrime(n): n in [1..250] | IsPrime(NthPrime(n)+26)];

Select[Prime[Range[200]], PrimeQ[# + 26] &]

A055009 Smallest composite number x such that sigma(x + prime(n)#) = sigma(x) + prime(n)#, where prime(n)# = A002110(n) and sigma is A000203.

Original entry on oeis.org

434, 104, 44, 176, 2924, 34256, 83509, 539081, 254963216, 14600541172, 201346999808

Offset: 1

Labos Elemer, May 31 2000

a(12) <= 14841476269604. a(13) <= 314064788156864. - Donovan Johnson, Mar 17 2013

			a(7) = 83509 = 37*37*61, sigma(83509)+510510 = 87234+510510 = sigma(83509+510510) = sigma(594019) = 597744.

The prime solutions for particular sigma(x+primorial) = sigma(x)+primorial equations are in A049481-A049485.

A000203, A002110, A015913-A015917, A049902-A049905, A049481-A049485.

a(n)=my(P=prod(i=1,n,prime(i)),x=4); while(isprime(x) || sigma(x+P) != sigma(x)+P, x++); x \\ Charles R Greathouse IV, Feb 14 2013

a(9)-a(10) from Donovan Johnson, Oct 15 2008

a(11) from Donovan Johnson, Mar 08 2013

A155760 Primes p such that p+30 and p+60 are prime.

Original entry on oeis.org

7, 11, 13, 23, 29, 37, 41, 43, 53, 67, 71, 79, 97, 107, 137, 151, 163, 167, 181, 197, 211, 233, 251, 277, 307, 337, 349, 359, 379, 389, 401, 419, 431, 449, 461, 541, 547, 557, 571, 587, 601, 613, 617, 631, 709, 727, 797, 823, 827, 877, 881, 907, 911, 937, 953

Offset: 1

Vincenzo Librandi, Jan 26 2009

Subsequence of A049481. - Zak Seidov, Apr 10 2015

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

[p: p in PrimesUpTo(1000) | IsPrime(p + 30) and IsPrime(p + 60)]; // Vincenzo Librandi, Oct 30 2012

A155760:=n->`if`(isprime(n) and isprime(n+30) and isprime(n+60),n,NULL): seq(A155760(n), n=1..2000); # Wesley Ivan Hurt, Apr 11 2015

Select[Prime[Range[1000]], PrimeQ[# +30] && PrimeQ[# + 60]&] (* Vincenzo Librandi, Oct 30 2012 *)

select(p->isprime(p+30)&&isprime(p+60), primes(10^3)) \\ Charles R Greathouse IV, Apr 11 2015

A282423 a(n) = smallest k such that A282026(k) = n, or 0 if no such k exists.

Original entry on oeis.org

3, 2, 0, 13, 19, 0, 427, 4, 0, 0, 1, 0, 802, 99412, 0, 3097, 7, 0, 637, 0, 0, 7225627, 30898822, 0, 0, 280134277, 0, 31705902442, 43190647, 0, 965577112

Offset: 1

Andrey Zabolotskiy and Altug Alkan, Feb 14 2017, following a suggestion from N. J. A. Sloane

a(n) is nonzero if n is in A282429.

For n>4 and nonzero a(n), 2*a(n)+3 is in A022004. For n>8 and nonzero a(n), 2*a(n)+3 is also in A153417. For n>16 and nonzero a(n), 2*a(n)+3 is also in A049481.

			a(10) = 0. Proof: Suppose 10 is a term of A282026. For the corresponding n, 2*n + 1 cannot be divisible by 5 because of A282026’s definition (gcd(10, 2*n + 1) = 1). So 2*n + 1 can be only of the form 10*k + 1, 10*k + 3, 10*k + 7, 10*k + 9. But 10*k + 1 + 2*2, 10*k + 3 + 2*1, 10*k + 7 + 2*4, 10*k + 9 + 2*8 are all composite and 1, 2, 4, 8 are relatively prime to any odd number. Since all of them are smaller than 10, this is the contradiction to the assumption that 10 is the term which is the smallest number for corresponding n. This also proves that a(5*k) = 0 for any k > 1.

Cf. A282026, A282429.

cp's OEIS Frontend

A156104 Primes p such that p+36 is also prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

A049482 Primes p such that p + 210 is also prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A049485 Primes p such that p + 510510 is also prime, where 510510 is the 7th primorial number A002110(7).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A154114 Primes p such that p + 9699690 is also prime, where 9699690 is the 8th primorial number A002110(8).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A049483 Primes p such that p + 2310 is also prime, where 2310 is the 5th primorial number A002110(5).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A049484 Primes p such that p + 30030 is also prime, where 30030 is the 6th primorial number A002110(6).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A252089 Primes p such that p + 26 is prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

A055009 Smallest composite number x such that sigma(x + prime(n)#) = sigma(x) + prime(n)#, where prime(n)# = A002110(n) and sigma is A000203.

Views

Author

Keywords

Comments