A102700 -id:A102700 - cp's OEIS frontend

A139367 a(n) = A102700(n) - A102342(n).

1, 1, 2, 3, 2, 1, 3, 2, 4, 4, 6, 4, 4, 9, 8, 7, 7, 7, 7, 7, 5, 2, 3, 2, 2, 4, 4, 7, 10, 10, 9, 9, 8, 7, 5, 6, 9, 7, 13, 13, 13, 11, 12, 14, 15, 23, 16, 16, 16, 10, 10, 16, 13, 13, 13, 13, 12, 7, 10, 7, 1, 1, 1, 4, 6, 5, 9, 9, 13, 11, 16, 13, 13, 15, 13, 8, 9, 8, 10, 9, 10, 6, 7, 7, 10, 10, 11, 18

Offset: 1

Juri-Stepan Gerasimov, Jun 09 2008

sign

a(332) = -1.

Index entries for prime races

From R. J. Mathar, Apr 25 2010: (Start)
A102700 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if isprime(10*a+9) then return a; end if; end do: end if; end proc:
A102342 := proc(n) option remember; if n = 1 then 0; else for a from procname(n-1)+1 do if isprime(10*a+7) then return a; end if; end do: end if; end proc:
A139367 := proc(n) A102700(n)-A102342(n) ; end proc: seq(A139367(n),n=1..120) ; (End)

Corrected (a 6 replaced by 7, a 7 by 9, a 13 by 16) and extended by R. J. Mathar, Apr 25 2010

A030433 Primes of form 10*k + 9.

Original entry on oeis.org

19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 349, 359, 379, 389, 409, 419, 439, 449, 479, 499, 509, 569, 599, 619, 659, 709, 719, 739, 769, 809, 829, 839, 859, 919, 929, 1009, 1019, 1039, 1049, 1069, 1109, 1129, 1229, 1249, 1259, 1279, 1289

Offset: 1

Warut Roonguthai

nonn

Also primes of form 5*k + 4.
5 is quadratic residue of primes of form 10*k-1. - Vincenzo Librandi, Jun 25 2014
Also, primes p such that 5 divides sigma(p), cf. A274397. - M. F. Hasler, Jul 10 2016
Conjecture: Primes p such that ((x+1)^5-1)/x has 2 distinct irreducible factors of degree 2 over GF(p). - Federico Provvedi, Apr 01 2018
The digital root of a(n) is 1, 2, 4, 5, 7 or 8. - Muniru A Asiru, Apr 28 2018
From Jianing Song, Sep 13 2022: (Start)
Primes p such that the ideal (p) factors into two prime ideals in Z[zeta_5], where zeta_5 = exp(2*Pi*i/5). Since Z[zeta_5] is a PID, this is equivalent to saying that this sequence lists primes p that are the product of two non-associate prime elements Z[zeta_5]. In particular, the factorization of p == 4 (mod 5) in Z[zeta_5] coincides with the factorization in Z[(1+sqrt(5))/2] (e.g., 19 = (8+3*sqrt(5))*(8-3*sqrt(5)) is the factorization of 19 in both Z[(1+sqrt(5))/2] and Z[zeta_5]).
Also primes p such that x^4 + x^3 + x^2 + x + 1 factors into two irreducible quadratic polynomials over GF(p) (cf. A327753). (End)

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.
Erika Klarreich, Mathematicians Discover Prime Conspiracy, Quanta Magazine, 2016.
R. J. Lemke Oliver and K. Soundararajan, Unexpected biases in the distribution of consecutive primes, arXiv:1603.03720 [math.NT], 2016.

Cf. A045457, A049510, A102700.

Filtered(List([1..500],n->10*n+9),IsPrime); # Muniru A Asiru, Apr 27 2018

select(isprime,[seq(10*n+9,n=1..500)]); # Muniru A Asiru, Apr 27 2018

Select[Prime@Range[210], Mod[ #, 10] == 9 &] (* Ray Chandler, Nov 07 2006 *)
Select[Range[9, 1300, 10], PrimeQ] (* Harvey P. Dale, Jun 01 2012 *)
Prime@Flatten@Position[Length@FactorList[((1+d)^5-1)/d,Modulus->#]&/@Prime@Range@200,3] (* Federico Provvedi, Apr 04 2018 *)

select(n->n%10==9, primes(100)) \\ Charles R Greathouse IV, Apr 29 2015

for(n=1, 1e3, if(isprime(p=10*n+9), print1(p, ", "))); \\ Altug Alkan, Apr 19 2018

a(n) = 10*A102700(n) + 9.
Union of A132234 and A132236. - Ray Chandler, Apr 07 2009
Intersection of A000040 and A017377. - Iain Fox, Dec 30 2017

Extended by Ray Chandler, Nov 07 2006

A007811 Numbers k for which 10k+1, 10k+3, 10k+7 and 10k+9 are primes.

Original entry on oeis.org

1, 10, 19, 82, 148, 187, 208, 325, 346, 565, 943, 1300, 1564, 1573, 1606, 1804, 1891, 1942, 2101, 2227, 2530, 3172, 3484, 4378, 5134, 5533, 6298, 6721, 6949, 7222, 7726, 7969, 8104, 8272, 8881, 9784, 9913, 10111, 10984, 11653, 11929, 12220, 13546, 14416, 15727

Offset: 1

N. J. A. Sloane and J. H. Conway, Mar 15 1996

nonn

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

Cf. A024912, A102338, A102342, A102700, A007530, A014561, A008471, A032352, A216292, A216293, A125855.

Cf. A010051, A245304, A245305.

a007811 n = a007811_list !! (n-1)
a007811_list = map (pred . head) $ filter (all (== 1) . map a010051') $
   iterate (zipWith (+) [10, 10, 10, 10]) [1, 3, 7, 9]
-- Reinhard Zumkeller, Jul 18 2014

[n: n in [0..10000] | forall{10*n+r: r in [1,3,7,9] | IsPrime(10*n+r)}]; // Bruno Berselli, Sep 04 2012

for n from 1 to 10000 do m := 10*n: if isprime(m+1) and isprime(m+3) and isprime(m+7) and isprime(m+9) then print(n); fi: od: quit

Select[ Range[ 1, 10000, 3 ], PrimeQ[ 10*#+1 ] && PrimeQ[ 10*#+3 ] && PrimeQ[ 10*#+7 ] && PrimeQ[ 10*#+9 ]& ]
Select[Range[15000], And @@ PrimeQ /@ ({1, 3, 7, 9} + 10#) &] (* Ray Chandler, Jan 12 2007 *)

p=2;q=3;r=5;forprime(s=7,1e5,if(s-p==8 && r-p==6 && q-p==2 && p%10==1, print1(p", ")); p=q;q=r;r=s) \\ Charles R Greathouse IV, Mar 21 2013

use ntheory ":all"; my @s = map { ($-1)/10 } sieve_prime_cluster(10,1e9, 2,6,8); say for @s; # _Dana Jacobsen, May 04 2017

a(n) = 3*A014561(n) + 1. - Zak Seidov, Sep 21 2009

A023240 Primes p such that 10*p + 9 is also prime.

Original entry on oeis.org

2, 5, 7, 13, 17, 19, 23, 37, 41, 43, 47, 59, 61, 71, 73, 83, 101, 103, 127, 131, 139, 149, 157, 197, 199, 223, 233, 239, 257, 269, 271, 281, 293, 307, 311, 331, 349, 353, 373, 401, 409, 421, 433, 463, 467, 479, 491, 499, 503, 509, 541, 547, 563, 577, 587, 593, 607, 619

Offset: 1

David W. Wilson

nonn

John Cerkan, Table of n, a(n) for n = 1..10000

Subsequence of A102700.

[ n: n in PrimesUpTo(700) | IsPrime(10*n+9) ]; // Vincenzo Librandi, Nov 20 2010

Select[Prime@ Range@ 120, PrimeQ[10 # + 9] &] (* Michael De Vlieger, Sep 12 2016 *)

A049510 Numbers k such that prime(k) == 9 (mod 10).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

nonn

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

Cf. A000720, A030433, A102700.

Cf. A049508, A049509, A049511.

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

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

Extended by Ray Chandler, Nov 07 2006

A023301 Primes that remain prime through 3 iterations of function f(x) = 10x + 9.

Original entry on oeis.org

13, 139, 293, 331, 547, 967, 1049, 1399, 1567, 1889, 1997, 2087, 2137, 2309, 2423, 2437, 2753, 2939, 3719, 3761, 3919, 4451, 4517, 4621, 6089, 7001, 7741, 8423, 8849, 9437, 10487, 11657, 12007, 12347, 12823, 13469, 15289, 15373, 15661, 17737, 17989

Offset: 1

David W. Wilson

nonn

Primes p such that 10*p+9, 100*p+99 and 1000*p+999 are also primes. - Vincenzo Librandi, Aug 04 2010

John Cerkan, Table of n, a(n) for n = 1..10000

Subsequence of A023240, A023270, and A102700.

[n: n in [1..450000] | IsPrime(n) and IsPrime(10*n+9) and IsPrime(100*n+99) and IsPrime(1000*n+999)] // Vincenzo Librandi, Aug 04 2010

nrp3Q[n_]:=AllTrue[Rest[NestList[10#+9&,n,3]],PrimeQ]; Select[Prime[ Range[ 2100]],nrp3Q] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 12 2019 *)

A104045 Numbers k such that k9 is prime and k is a multiple of ten.

Original entry on oeis.org

10, 40, 50, 70, 80, 100, 110, 140, 160, 170, 230, 260, 290, 310, 320, 370, 440, 490, 500, 520, 530, 670, 710, 730, 800, 820, 860, 910, 920, 1000, 1070, 1090, 1190, 1210, 1240, 1280, 1300, 1310, 1330, 1370, 1400, 1580, 1720, 1750, 1760, 1790, 1900, 1930, 1960, 1970, 2050, 2080, 2210

Offset: 1

Parthasarathy Nambi, Mar 01 2005

			If k =  10, then k9 =  109 (prime).
If k = 160, then k9 = 1609 (prime).
If k = 320, then k9 = 3209 (prime).

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A030433, A008592, A102700, A166560 (resulting primes).

select(n-> isprime(10*n+9), [10*i$i=1..300])[];  # Alois P. Heinz, Jan 19 2024

Select[Range[10,2210,10],PrimeQ[FromDigits[Prepend[{9},#]]]&] (* James C. McMahon, Jan 19 2024 *)

from sympy import isprime
from itertools import count, islice
def agen(): yield from (k for k in count(10, 10) if isprime(10*k+9))
print(list(islice(agen(), 53))) # Michael S. Branicky, Jan 19 2024

A023329 Primes that remain prime through 4 iterations of function f(x) = 10x + 9.

Original entry on oeis.org

13, 139, 293, 1889, 2939, 3719, 6089, 7741, 12823, 19753, 21391, 22861, 28513, 36721, 37967, 40949, 60899, 76519, 83621, 101747, 121687, 127549, 128239, 142099, 149197, 153817, 155581, 158489, 160159, 169283, 173651, 180749, 185831, 192037, 198221

Offset: 1

David W. Wilson

nonn

Primes p such that 10*p+9, 100*p+99, 1000*p+999 and 10000*p+9999 are also primes. - Vincenzo Librandi, Aug 04 2010

John Cerkan, Table of n, a(n) for n = 1..10000

Subsequence of A023240, A023270, A023301, and A102700.

[n: n in [1..5000000] | IsPrime(n) and IsPrime(10*n+9) and IsPrime(100*n+99) and IsPrime(1000*n+999) and IsPrime(10000*n+9999)]; // Vincenzo Librandi, Aug 04 2010

Select[Prime[Range[20000]],AllTrue[Rest[NestList[10#+9&,#,4]],PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 16 2020 *)

a(n) == 9 or 13 (mod 14). - John Cerkan, Oct 09 2016

Definition clarified by Harvey P. Dale, Feb 16 2020

A153380 Numbers n such that 10*n+9 is not prime.

Original entry on oeis.org

0, 3, 4, 6, 9, 11, 12, 15, 16, 18, 20, 21, 24, 25, 27, 28, 29, 30, 31, 32, 33, 36, 39, 42, 45, 46, 48, 51, 52, 53, 54, 55, 57, 58, 60, 62, 63, 64, 66, 67, 68, 69, 72, 74, 75, 77, 78, 79, 81, 84, 86, 87, 88, 89, 90, 93, 94, 95, 96, 97, 98, 99, 102

Offset: 1

Vincenzo Librandi, Dec 25 2008

			Triangle begins:
0;
*,*;
*,*,4;
*,*,*,*;
*,*,*,*,*;
3,*,*,*,*,16;
*,*,*,*,*,*,*;
*,*,11,*,*,*,*,28;
*,*,*,*,20,*,*,*,*;
*,*,*,*,*,*,*,*,*,*;
6,*,*,*,*,29,*,*,*,*,52;
where * marks the non-integer values of (2*h*k + k + h - 4)/5 with h >= k >= 1. - _Vincenzo Librandi_, Jan 13 2013

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

Cf. A102700.

[n: n in [0..150] | not IsPrime(10*n+9)]; // Vincenzo Librandi, Jan 13 2013

Select[Range[0, 200], !PrimeQ[10 # + 9] &] (* Vincenzo Librandi, Jan 13 2013 *)

0 added by Arkadiusz Wesolowski, Aug 03 2011

cp's OEIS Frontend

A139367 a(n) = A102700(n) - A102342(n).

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A030433 Primes of form 10*k + 9.

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A007811 Numbers k for which 10k+1, 10k+3, 10k+7 and 10k+9 are primes.

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A023240 Primes p such that 10*p + 9 is also prime.

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A049510 Numbers k such that prime(k) == 9 (mod 10).

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A023301 Primes that remain prime through 3 iterations of function f(x) = 10x + 9.

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Magma

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A104045 Numbers k such that k9 is prime and k is a multiple of ten.

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A023329 Primes that remain prime through 4 iterations of function f(x) = 10x + 9.