A042993 -id:A042993 - cp's OEIS frontend

A167134 Primes congruent to {2, 3, 5, 7} mod 11.

2, 3, 5, 7, 13, 29, 47, 71, 73, 79, 101, 113, 137, 139, 157, 167, 179, 181, 211, 223, 227, 233, 269, 271, 277, 293, 311, 313, 337, 359, 379, 401, 409, 421, 431, 443, 467, 487, 491, 509, 541, 557, 563, 577, 599, 601, 607, 619, 641, 643, 673, 709, 733, 739, 751

Offset: 1

Klaus Brockhaus, Oct 28 2009

Primes p such that p mod 11 is prime.
Primes of the form 11*n+r where n >= 0 and r is in {2, 3, 5, 7}.
2 and primes congruent to {3, 5, 7, 13} mod 22. - Chai Wah Wu, Apr 29 2025

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

Cf. A003627, A045326, A003631, A045309, A045314, A042987, A078403, A042993, A167134, A167135, A167119: primes p such that p mod k is prime, for k = 3..13 resp.

[ p: p in PrimesUpTo(760) | p mod 11 in {2, 3, 5, 7} ];
[ p: p in PrimesUpTo(760) | exists(t){ n: n in [0..p div 11] | exists(u){ r: r in {2, 3, 5,7} | p eq (11*n+r) } } ];

Select[Prime[Range[600]],MemberQ[{2, 3, 5, 7},Mod[#,11]]&] (* Vincenzo Librandi, Aug 05 2012 *)

A167135 Primes congruent to {2, 3, 5, 7, 11} mod 12.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 19, 23, 29, 31, 41, 43, 47, 53, 59, 67, 71, 79, 83, 89, 101, 103, 107, 113, 127, 131, 137, 139, 149, 151, 163, 167, 173, 179, 191, 197, 199, 211, 223, 227, 233, 239, 251, 257, 263, 269, 271, 281, 283, 293, 307, 311, 317, 331, 347, 353, 359

Offset: 1

Klaus Brockhaus, Oct 28 2009

Primes p such that p mod 12 is prime.
Primes of the form 12*n+r where n >= 0 and r is in {2, 3, 5, 7, 11}.
Except for the prime 2, these are the primes that are encountered in the set of numbers {x, f(f(x))} where x is of the form 4k+3 with k>=0, and where f(x) is the 3x+1-problem function, and f(f(x)) the second iteration value. Indeed this sequence is the set union of 2 and A002145 (4k+3 primes) and A007528 (6k+5 primes), since f(f(4k+3))=6k+5. Equivalently one does not get any prime from A068228 (the complement of the present sequence). - Michel Marcus and Bill McEachen, May 07 2016

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

Subsequences: A002145, A007528. Complement: A068228.

Cf. A003627, A045326, A003631, A045309, A045314, A042987, A078403, A042993, A167134, A167135, A167119: primes p such that p mod k is prime, for k = 3..13 resp.

[ p: p in PrimesUpTo(760) | p mod 12 in {2, 3, 5, 7, 11} ];

[ p: p in PrimesUpTo(760) | exists(t){ n: n in [0..p div 12] | exists(u){ r: r in {2, 3, 5,7, 11} | p eq (12*n+r) } } ];

isA167135  := n -> isprime(n) and not modp(n, 12) != 1:
select(isA167135, [$1..360]); # Peter Luschny, Mar 28 2018

Select[Prime[Range[400]],MemberQ[{2,3, 5, 7, 11},Mod[#,12]]&] (* Vincenzo Librandi, Aug 05 2012 *)
Select[Prime[Range[72]], Mod[#, 12] != 1 &] (* Peter Luschny, Mar 28 2018 *)

A201717 Primes of the form 3*m^2 - 5.

Original entry on oeis.org

7, 43, 103, 967, 1447, 1723, 2347, 3067, 3463, 4327, 6343, 6907, 9403, 11527, 13063, 21163, 23227, 28807, 32443, 33703, 44647, 47623, 52267, 65707, 71143, 74887, 80683, 88747, 90823, 99367, 110587, 137383, 142567, 150523, 175687

Offset: 1

Vincenzo Librandi, Dec 05 2011

Primes p such that 3*(p+5) or (p+5)/3 is a square. - Vincenzo Librandi, Feb 16 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A000040, A042993 (supersequence).

Cf. similar sequences: A089682, A201715, A201716, A201718, A201781.

[a: n in [2..300] | IsPrime(a) where a is 3*n^2-5];

Mathematica
```
Select[Table[3n^2-5,{n,2,1000}],PrimeQ]
```

A167119 Primes congruent to 2, 3, 5, 7 or 11 (mod 13).

Original entry on oeis.org

2, 3, 5, 7, 11, 29, 31, 37, 41, 59, 67, 83, 89, 107, 109, 137, 163, 167, 193, 197, 211, 223, 239, 241, 263, 271, 293, 317, 349, 353, 367, 379, 397, 401, 419, 421, 431, 449, 457, 479, 499, 509, 523, 557, 577, 587, 601, 613, 631, 653, 661, 683, 691, 709, 733, 739, 743, 757

Offset: 1

Vladimir Joseph Stephan Orlovsky, Oct 27 2009

Primes which have a remainder mod 13 that is prime.

Union of A141858, A100202, A102732, A140371 and A140373. - R. J. Mathar, Oct 29 2009

			11 mod 13 = 11, 29 mod 13 = 3, 31 mod 13 = 5, hence 11, 29 and 31 are in the sequence.

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

Cf. A003627, A045326, A003631, A045309, A045314, A042987, A078403, A042993, A167134, A167135: primes p such that p mod k is prime, for k = 3..12 resp.

[ p: p in PrimesUpTo(740) | p mod 13 in {2, 3, 5, 7, 11} ]; // Klaus Brockhaus, Oct 28 2009

f[n_]:=PrimeQ[Mod[n,13]]; lst={};Do[p=Prime[n];If[f[p],AppendTo[lst,p]],{n,6,6!}];lst
Select[Prime[Range[4000]],MemberQ[{2, 3, 5, 7, 11},Mod[#,13]]&] (* Vincenzo Librandi, Aug 05 2012 *)

{forprime(p=2, 740, if(isprime(p%13), print1(p, ",")))} \\ Klaus Brockhaus, Oct 28 2009

Edited by Klaus Brockhaus and R. J. Mathar, Oct 28 2009 and Oct 29 2009

A260181 Numbers whose last digit is prime.

Original entry on oeis.org

2, 3, 5, 7, 12, 13, 15, 17, 22, 23, 25, 27, 32, 33, 35, 37, 42, 43, 45, 47, 52, 53, 55, 57, 62, 63, 65, 67, 72, 73, 75, 77, 82, 83, 85, 87, 92, 93, 95, 97, 102, 103, 105, 107, 112, 113, 115, 117, 122, 123, 125, 127, 132, 133, 135, 137, 142, 143, 145, 147

Offset: 1

Wesley Ivan Hurt, Jul 17 2015

Numbers ending in 2, 3, 5 or 7.
The subsequence of primes is A042993. - Michel Marcus, Jul 19 2015
From Wesley Ivan Hurt, Aug 15 2015, Sep 26 2015: (Start)
Ceiling(a(n)/2) = A047201(n).
Complement of (A197652 Union A262389). (End)

Muniru A Asiru, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A042993, A047201, A092620, subset of A118950.
Union of A017293, A017305, A017329 and A017353.
First differences are [1,2,2,5,...] = A002522(A140081(n-1)).
Cf. A197652, A262389.

a:=n->(5*n-4-(-1)^n+((3-(-1)^n)/2)*(-1)^((2*n+5-(-1)^n)/4))/2; List([1..60],n->a(n)); # Muniru A Asiru, Feb 16 2018

[(5*n-4-(-1)^n+((3-(-1)^n) div 2)*(-1)^((2*n+5-(-1)^n) div 4))/2: n in [1..70]]; // Vincenzo Librandi, Jul 18 2015

A260181:=n->(5*n-4-(-1)^n+((3-(-1)^n)/2)*(-1)^((2*n+5-(-1)^n)/4))/2: seq(A260181(n), n=1..100);

CoefficientList[Series[(2 + x + 2 x^2 + 2 x^3 + 3 x^4)/((x - 1)^2*(1 + x + x^2 + x^3)), {x, 0, 100}], x]
LinearRecurrence[{1, 0, 0, 1, -1}, {2, 3, 5, 7, 12}, 60] (* Vincenzo Librandi, Jul 18 2015 *)
Table[(5n - 4 - (-1)^n + ((3 - (-1)^n)/2)*(-1)^((2*n + 5 - (-1)^n)/4))/2, {n, 100}] (* Wesley Ivan Hurt, Aug 11 2015 *)

is(n)=my(m=digits(n));isprime(m[#m]) \\ Anders Hellström, Jul 19 2015

A260181(n)=(n--)\4*10+prime(n%4+1) \\ is(n)=isprime(n%10) is much more efficient than the above. - M. F. Hasler, Sep 16 2016

G.f.: x*(2+x+2*x^2+2*x^3+3*x^4) / ((x-1)^2*(1+x+x^2+x^3)).
a(n) = a(n-1)+a(n-4)-a(n-5), n>5.
a(n) = (5*n-4-(-1)^n+((3-(-1)^n)/2)*(-1)^((2*n+5-(-1)^n)/4))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(5*sqrt(5+2*sqrt(5))) - 25*log(5) - 40*log(2) + 5*sqrt(5)*arccoth(843/2))/200. - Amiram Eldar, Jul 30 2024

A002382 Numbers of the form (p^2 - 49)/120 where p is prime.

Original entry on oeis.org

0, 1, 2, 4, 11, 15, 18, 23, 37, 44, 57, 78, 88, 95, 106, 134, 156, 205, 221, 232, 249, 310, 323, 414, 429, 452, 550, 576, 639, 667, 715, 785, 816, 837, 946, 1003, 1038, 1122, 1159, 1222, 1313, 1562, 1635, 1740, 1786, 1817, 1976, 2108, 2279, 2493, 2585, 2641

Offset: 1

N. J. A. Sloane

nonn

Primes p corresponding to a(n) are found in A003631(n+2) = A042993(n+3) = A097957(n+1). - Ray Chandler, Jul 29 2019

H. Gupta, On a conjecture of Chowla, Proc. Indian Acad. Sci., 5A (1937), 381-384.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)
H. Gupta, On a conjecture of Chowla, Proc. Indian Acad. Sci., 5A (1937), 381-384. [Annotated scanned copy]

Cf. A002381, A002855, A003631, A042993, A097957.

Select[(Prime[Range[150]]^2-49)/120,IntegerQ] (* Harvey P. Dale, Jan 19 2014 *)

More terms from James Sellers, May 03 2000

A215374 Primes congruent to {0, 2, 3} mod 11.

Original entry on oeis.org

2, 3, 11, 13, 47, 79, 101, 113, 157, 167, 179, 211, 223, 233, 277, 311, 409, 421, 431, 443, 487, 509, 541, 563, 607, 619, 641, 673, 739, 751, 761, 773, 827, 839, 883, 937, 971, 1069, 1091, 1103, 1201, 1213, 1223, 1279, 1289, 1301, 1367, 1399, 1433, 1487

Offset: 1

Vincenzo Librandi, Aug 09 2012

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

Cf. A000040, A042993, A045333.

[p: p in PrimesUpTo(1500) | p mod 11 in [0, 2, 3]];

Select[Prime[Range[400]],MemberQ[{0,2,3},Mod[#,11]]&]

A215375 Primes congruent to {0, 2, 3} mod 13.

Original entry on oeis.org

2, 3, 13, 29, 41, 67, 107, 197, 211, 223, 263, 353, 367, 379, 419, 431, 457, 509, 523, 587, 601, 613, 653, 691, 743, 757, 769, 809, 821, 887, 977, 991, 1069, 1237, 1277, 1289, 1303, 1367, 1381, 1433, 1459, 1471, 1511, 1523, 1549, 1601, 1627, 1667, 1693, 1783

Offset: 1

Vincenzo Librandi, Aug 09 2012

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

Cf. A000040, A042993, A045333.

[p: p in PrimesUpTo(2000) | p mod 13 in [0, 2, 3]];

Select[Prime[Range[400]],MemberQ[{0,2,3},Mod[#,13]]&]

A215376 Primes congruent to {0, 2, 3} mod 17.

Original entry on oeis.org

2, 3, 17, 19, 37, 53, 71, 139, 173, 223, 241, 257, 359, 461, 479, 547, 563, 631, 683, 733, 751, 853, 887, 937, 971, 1039, 1091, 1193, 1277, 1447, 1481, 1499, 1549, 1567, 1583, 1601, 1669, 1753, 1787, 1873, 1889, 1907, 2111, 2161, 2179, 2213, 2281, 2297

Offset: 1

Vincenzo Librandi, Aug 09 2012

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

Cf. A000040, A042993, A045333.

[p: p in PrimesUpTo(2500) | p mod 17 in [0, 2, 3]];

Select[Prime[Range[500]],MemberQ[{0,2,3},Mod[#,17]]&]

A215377 Primes congruent to {0, 2, 3} mod 19.

Original entry on oeis.org

2, 3, 19, 41, 59, 79, 97, 173, 193, 211, 269, 307, 383, 401, 421, 439, 743, 839, 857, 877, 953, 971, 991, 1009, 1123, 1181, 1237, 1409, 1427, 1447, 1523, 1579, 1637, 1693, 1789, 1979, 1997, 2017, 2111, 2131, 2207, 2339, 2377, 2473

Offset: 1

Vincenzo Librandi, Aug 09 2012

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

Cf. A000040, A042993, A045333.

[p: p in PrimesUpTo(2500) | p mod 19 in [0, 2, 3]];

Select[Prime[Range[500]],MemberQ[{0,2,3},Mod[#,19]]&]

cp's OEIS Frontend

A167134 Primes congruent to {2, 3, 5, 7} mod 11.

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A167135 Primes congruent to {2, 3, 5, 7, 11} mod 12.

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A201717 Primes of the form 3*m^2 - 5.

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A167119 Primes congruent to 2, 3, 5, 7 or 11 (mod 13).

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A260181 Numbers whose last digit is prime.

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A002382 Numbers of the form (p^2 - 49)/120 where p is prime.

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A215374 Primes congruent to {0, 2, 3} mod 11.

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Magma