A049508 -id:A049508

A030431

Primes of form 10n+3.

+10
48

3, 13, 23, 43, 53, 73, 83, 103, 113, 163, 173, 193, 223, 233, 263, 283, 293, 313, 353, 373, 383, 433, 443, 463, 503, 523, 563, 593, 613, 643, 653, 673, 683, 733, 743, 773, 823, 853, 863, 883, 953, 983, 1013, 1033, 1063, 1093, 1103, 1123, 1153, 1163, 1193

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Also primes of form 5n+3.

Union of A132233, A132235, {3}. - Ray Chandler, Apr 07 2009

Primes p such that arithmetic mean of divisors of p^4 is an integer. There are 2 such sequences of primes, this one and A030430. - Ctibor O. Zizka, Oct 20 2009

5 is not quadratic residue of primes of this form. - Vincenzo Librandi, Jun 25 2014

Intersection of A000040 and A017305. - Iain Fox, Dec 30 2017

LINKS

Michael B. Porter, Table of n, a(n) for n = 1..100000

A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.

FORMULA

a(n) = 10*A102338(n) + 3.

MATHEMATICA

Select[Prime@Range[200], Mod[ #, 10] == 3 &] (* Ray Chandler, Nov 07 2006 *)

Select[10 Range[0, 150] + 3, PrimeQ] (* Harvey P. Dale, Apr 06 2011 *)

PROG

(PARI) select(n->n%10==3, primes(500)) \\ Charles R Greathouse IV, Apr 29 2015

CROSSREFS

Cf. A045414, A049508, A053179, A102338.

KEYWORD

nonn

AUTHOR

Warut Roonguthai

EXTENSIONS

Extended by Ray Chandler, Nov 07 2006

STATUS

approved

A049511

Numbers k such that prime(k) == 1 (mod 10).

+10
9

5, 11, 13, 18, 20, 26, 32, 36, 42, 43, 47, 53, 54, 58, 60, 64, 67, 79, 82, 83, 89, 94, 98, 100, 105, 110, 115, 116, 121, 125, 126, 133, 135, 141, 142, 152, 156, 160, 164, 167, 172, 173, 177, 178, 182, 190, 193, 194, 197, 202, 210, 212, 216, 218, 221, 230, 233

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Also k for which prime(k) == 1 (mod 5). - Bruno Berselli, Mar 04 2016

The asymptotic density of this sequence is 1/4 (by Dirichlet's theorem). - Amiram Eldar, Mar 01 2021

LINKS

Zak Seidov, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = A000720(A030430(n)). - Ray Chandler, Nov 07 2006

MATHEMATICA

Select[Range[210], Mod[Prime[ # ], 10] == 1 &] (* Ray Chandler, Nov 07 2006 *)

PROG

(Sage) [n for n in (1..300) if Mod(nth_prime(n), 10) == 1] # Bruno Berselli, Mar 04 2016

(PARI) isok(n) = !((prime(n)-1) % 10); \\ Michel Marcus, Mar 04 2016

CROSSREFS

Cf. A000720, A024912, A030430, A081759.

Cf. A049508, A049509, A049510.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

Extended by Ray Chandler, Nov 28 2003

Formula corrected by Zak Seidov, Sep 20 2011

STATUS

approved

A102338

Numbers k such that 10k+3 is prime.

+10
9

0, 1, 2, 4, 5, 7, 8, 10, 11, 16, 17, 19, 22, 23, 26, 28, 29, 31, 35, 37, 38, 43, 44, 46, 50, 52, 56, 59, 61, 64, 65, 67, 68, 73, 74, 77, 82, 85, 86, 88, 95, 98, 101, 103, 106, 109, 110, 112, 115, 116, 119, 121, 122, 128, 130, 137, 142, 143, 145, 148, 149, 152, 154, 155

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

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

EXAMPLE

For n=1, 10k+3 = 13 (prime).

For n=26, 10k+3 = 263 (prime).

For n=50, 10k+3 = 503 (prime).

MATHEMATICA

Select[Range[0, 160], PrimeQ[10# + 3] &] (* Ray Chandler, Nov 07 2006 *)

PROG

(Magma) [n: n in [0..1000]| IsPrime(10*n+3)]; // Vincenzo Librandi, Apr 06 2011

(PARI) isok(n) = isprime(10*n+3); \\ Michel Marcus, Sep 08 2016

CROSSREFS

Cf. A023238 (subsequence of primes), A030431, A049508.

KEYWORD

nonn

AUTHOR

Parthasarathy Nambi, Feb 20 2005

EXTENSIONS

Edited and extended by Ray Chandler, Nov 07 2006

STATUS

approved

A049509

Numbers k such that prime(k) == 7 (mod 10).

+10
6

4, 7, 12, 15, 19, 25, 28, 31, 33, 37, 39, 45, 49, 55, 59, 63, 66, 68, 69, 73, 78, 88, 91, 93, 101, 102, 106, 107, 111, 113, 118, 123, 129, 134, 138, 139, 144, 148, 151, 154, 155, 159, 161, 163, 165, 168, 181, 184, 187, 195, 199, 203, 206, 211, 214, 217, 219, 225

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The asymptotic density of this sequence is 1/4 (by Dirichlet's theorem). - Amiram Eldar, Mar 01 2021

LINKS

Table of n, a(n) for n=1..58.

FORMULA

a(n) = A000720(A030432(n)). - Ray Chandler, Nov 07 2006

MATHEMATICA

Select[Range[240], Mod[Prime[ # ], 10] == 7 &] (* Ray Chandler, Nov 07 2006 *)

CROSSREFS

Cf. A000720, A030432, A102342.

Cf. A049508, A049510, A049511.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

Extended by Ray Chandler, Nov 07 2006

STATUS

approved

A049510

Numbers k such that prime(k) == 9 (mod 10).

+10
5

8, 10, 17, 22, 24, 29, 34, 35, 41, 46, 50, 52, 57, 70, 72, 75, 77, 80, 81, 85, 87, 92, 95, 97, 104, 109, 114, 120, 127, 128, 131, 136, 140, 145, 146, 149, 157, 158, 169, 171, 175, 176, 180, 186, 189, 201, 204, 205, 207, 209, 215, 222, 223, 226, 228, 232, 237, 239

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The asymptotic density of this sequence is 1/4 (by Dirichlet's theorem). - Amiram Eldar, Mar 01 2021

LINKS

Table of n, a(n) for n=1..58.

FORMULA

a(n) = A000720(A030433(n)). - Ray Chandler, Nov 07 2006

MATHEMATICA

Select[Range[240], Mod[Prime[ # ], 10] == 9 &] (* Ray Chandler, Nov 07 2006 *)

CROSSREFS

Cf. A000720, A030433, A102700.

Cf. A049508, A049509, A049511.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

Extended by Ray Chandler, Nov 07 2006

STATUS

approved

A244739

Numbers k such that (prime(k) mod 5) == 0 (mod 3).

+10
4

2, 3, 6, 9, 14, 16, 21, 23, 27, 30, 38, 40, 44, 48, 51, 56, 61, 62, 65, 71, 74, 76, 84, 86, 90, 96, 99, 103, 108, 112, 117, 119, 122, 124, 130, 132, 137, 143, 147, 150, 153, 162, 166, 170, 174, 179, 183, 185, 188, 191, 192, 196, 198, 200, 208, 213, 220, 224

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Every positive integer is in exactly one of the sequences A244739, A024707, A244741.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..1000

EXAMPLE

n ... prime(n) mod 5 mod 3

1 ..... 2 ..... 2 ... 2

2 ..... 3 ..... 3 ... 0

3 ..... 5 ..... 0 ... 0

4 ..... 7 ..... 2 ... 2

5 ..... 11 .... 1 ... 1

6 ..... 13 .... 3 ... 0

MATHEMATICA

z = 300; u = Mod[Table[Mod[Prime[n], 5], {n, 1, z}], 3] (* A244738 *)

v1 = Flatten[Position[u, 0]] (* A244739 *)

v2 = Flatten[Position[u, 1]] (* A024707 *)

v3 = Flatten[Position[u, 2]] (* A244741 *)

CROSSREFS

Cf. A039703, A244738, A024707, A244741, A244735. Essentially the same as A049508.

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jul 05 2014

STATUS

approved

A240839

Both n and prime(n) are primes congruent to 3 (mod 10).

+10
1

23, 103, 293, 503, 823, 883, 953, 983, 1033, 1163, 1213, 1223, 1433, 1453, 1493, 1523, 1723, 1733, 1933, 1993, 2113, 2203, 2803, 2833, 2903, 3023, 3203, 3343, 3433, 3733, 3823, 3833, 4003, 4243, 4373, 4483, 4513, 4733, 4813, 4903, 4943, 4993, 5333, 5503, 5743, 6143, 6343, 6833, 7013

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Intersection of A030431 and A049508.

LINKS

Zak Seidov, Table of n, a(n) for n = 1..469

EXAMPLE

prime(23, 103, 293, 503, 823, 883, 953, 983, 1033, 1163) = (83, 563, 1913, 3593, 6323, 6863, 7523, 7753, 8233, 9403).

MATHEMATICA

Intersection[A030431 = Select[Range[3, 1000003, 10], PrimeQ], PrimePi[A030431]] (* gives 469 terms for prime(n) up to 10^6 *)

Select[Prime[Range[50000]], Mod[#, 10]==Mod[Prime[#], 10]==3&] (* gives 3126 terms from the first 50000 primes *)(* Harvey P. Dale, Nov 29 2014 *)

PROG

(PARI) s=[]; forprime(n=2, 8000, if(n%10==3 && prime(n)%10==3, s=concat(s, n))); s \\ Colin Barker, Apr 16 2014

CROSSREFS

Cf. A030431, A049508.

KEYWORD

nonn

AUTHOR

Zak Seidov, Apr 13 2014

STATUS

approved