A024912 -id:A024912 - cp's OEIS frontend

A139368 a(n) = A024912(n) - A102342(n).

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

Offset: 1

Juri-Stepan Gerasimov, Jun 09 2008

sign

A139368(219) = -4.

Index entries for prime races

From R. J. Mathar, Apr 25 2010: (Start)
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:
A024912 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if isprime(10*a+1) then return a; end if; end do: end if; end proc:
A139368 := proc(n) A024912(n)-A102342(n) ; end proc: seq(A139368(n),n=1..120) ; (End)

More terms from R. J. Mathar, Apr 25 2010

A030430 Primes of the form 10*n+1.

Original entry on oeis.org

11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251, 271, 281, 311, 331, 401, 421, 431, 461, 491, 521, 541, 571, 601, 631, 641, 661, 691, 701, 751, 761, 811, 821, 881, 911, 941, 971, 991, 1021, 1031, 1051, 1061, 1091, 1151, 1171, 1181, 1201, 1231, 1291

Offset: 1

Warut Roonguthai

Also primes of form 5*n+1 or equivalently 5*n+6.
Primes p such that the arithmetic mean of divisors of p^4 is an integer: A000203(p^4)/A000005(p^4) = C. - Ctibor O. Zizka, Sep 15 2008
Being a subset of A141158, this is also a subset of the primes of form x^2-5*y^2. - Tito Piezas III, Dec 28 2008
5 is quadratic residue of primes of this form. - Vincenzo Librandi, Jun 25 2014
Primes p such that 5 divides sigma(p^4), cf. A274397. - M. F. Hasler, Jul 10 2016

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.
Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 519.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.

Cf. A024912, A045453, A049511, A081759, A017281, A010051, A004615 (multiplicative closure).
Cf. A001583 (subsequence).
Union of A132230 and A132232. - Ray Chandler, Apr 07 2009

a030430 n = a030430_list !! (n-1)
a030430_list = filter ((== 1) . a010051) a017281_list
-- Reinhard Zumkeller, Apr 16 2012

Select[Prime@Range[210], Mod[ #, 10] == 1 &] (* Ray Chandler, Dec 06 2006 *)
Select[Range[11,1291,10],PrimeQ] (*Zak Seidov, Aug 14 2011*)

is(n)=n%10==1 && isprime(n) \\ Charles R Greathouse IV, Sep 06 2012

lista(nn) = forprime(p=11, nn, if(p%10==1, print1(p, ", "))) \\ Iain Fox, Dec 30 2017

a(n) = 10*A024912(n)+1 = 5*A081759(n)+6.
A104146(floor(a(n)/10)) = 1.
Union of A132230 and A132232. - Ray Chandler, Apr 07 2009
a(n) ~ 4n log n. - Charles R Greathouse IV, Sep 06 2012
Intersection of A000040 and A017281. - Iain Fox, Dec 30 2017

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

A081759 Numbers n such that 5n+6 is prime.

Original entry on oeis.org

1, 5, 7, 11, 13, 19, 25, 29, 35, 37, 41, 47, 49, 53, 55, 61, 65, 79, 83, 85, 91, 97, 103, 107, 113, 119, 125, 127, 131, 137, 139, 149, 151, 161, 163, 175, 181, 187, 193, 197, 203, 205, 209, 211, 217, 229, 233, 235, 239, 245, 257, 259, 263, 271, 275, 289

Offset: 1

Giovanni Teofilatto, Nov 21 2003

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

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A024912, A030430, A049511, A067076, A087505, A089192, A089253.

[n: n in [0..300]| IsPrime(5*n + 6)]; // Vincenzo Librandi, Oct 16 2012

A081759 := proc(n) option remember: local k: if(n=1)then return 1: fi: for k from procname(n-1)+1 do if(isprime(5*k+6))then return k: fi: od: end: seq(A081759(n),n=1..100); # Nathaniel Johnston, May 28 2011

Select[Range[300], PrimeQ[5# + 6] &] (* Ray Chandler, Dec 06 2006 *)

is(n)=isprime(5*n+6) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = 2*A024912(n) - 1.

Corrected by Ray Chandler, Nov 22 2003

A049511 Numbers k such that prime(k) == 1 (mod 10).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

nonn

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

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

Cf. A000720, A024912, A030430, A081759.

Cf. A049508, A049509, A049510.

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

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

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

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

Extended by Ray Chandler, Nov 28 2003

Formula corrected by Zak Seidov, Sep 20 2011

A023237 Primes p such that 10*p + 1 is also prime.

Original entry on oeis.org

3, 7, 13, 19, 31, 43, 97, 103, 109, 151, 157, 181, 193, 211, 241, 271, 337, 349, 367, 409, 421, 439, 487, 523, 547, 571, 601, 613, 631, 691, 733, 769, 811, 823, 829, 883, 937, 1009, 1021, 1033, 1039, 1063, 1069, 1117, 1201, 1249, 1279, 1291, 1459, 1483, 1489

Offset: 1

David W. Wilson

Primes which with a 1 appended stay prime.
Corresponding values of 10n + 1 in A055781. - Jaroslav Krizek, Jul 14 2010
Subsequence of A024912. - Michel Marcus, May 21 2014

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

Cf. A024912, A055781, A105435, A005384 (2*p + 1), A158017 (10*p - 1).

[n: n in [0..1000] | IsPrime(n) and IsPrime(10*n+1)]; // Vincenzo Librandi, Nov 20 2010

[p: p in PrimesUpTo(1100)| IsPrime(10*p+1)]; // Vincenzo Librandi, May 21 2014

with(numtheory); for i from 1 to 500 do if isprime(10*ithprime(i)+1) then printf(`%d,`, ithprime(i)) fi: od: # James Sellers, Apr 09 2005

Select[Prime[Range[ 240]], PrimeQ[FromDigits[Join[IntegerDigits[ # ], {1}]]] &] (* Robert G. Wilson v, Apr 09 2005 *)
Select[Prime[Range[900]], PrimeQ[10 # + 1] &] (* Vincenzo Librandi, May 21 2014 *)

Edited by N. J. A. Sloane at the suggestion of M. F. Hasler, Aug 24 2007

A124410 Numbers k such that 2k+1, 4k+1, 6k+1, 8k+1 and 10k+1 are primes.

Original entry on oeis.org

5415, 12705, 13020, 44370, 82950, 98280, 105525, 112200, 115140, 123855, 134250, 134460, 187740, 188745, 210165, 225705, 247170, 256410, 296310, 302085, 367875, 375645, 382890, 399585, 404040, 476340, 487830, 526845, 532095, 566430, 578085

Offset: 1

Artur Jasinski, Oct 31 2006

nonn

Jinyuan Wang, Table of n, a(n) for n = 1..1000

Cf. A005097, A005098, A024899, A005123, A024912, A123998, A124408, A124409, A124411, A071576.

Select[Range[600000], And @@ PrimeQ /@ ({2, 4, 6, 8, 10}*# + 1) &] (* Ray Chandler, Nov 20 2006 *)

is(k) = sum(j = 1, 5, isprime(2*j*k+1)) == 5; \\ Jinyuan Wang, Aug 04 2019

A124411 Numbers k such that 2k+1, 4k+1, 6k+1, 8k+1, 10k+1 and 12k+1 are primes.

Original entry on oeis.org

12705, 13020, 105525, 256410, 966840, 1707510, 1944495, 2310000, 2478630, 3132675, 3836070, 3976770, 4112430, 4532325, 5499585, 5920005, 6610485, 7390845, 8552250, 10739505, 11120340, 12231450, 12338130, 13243230, 16467255

Offset: 1

Artur Jasinski, Oct 31 2006

nonn

Jinyuan Wang, Table of n, a(n) for n = 1..100

Cf. A005097, A005098, A024899, A005123, A024912, A110801, A123998, A124408, A124409, A124410, A071576.

Select[Range[10^7], And @@ PrimeQ /@ ({2, 4, 6, 8, 10, 12}*# + 1) &] (* Ray Chandler, Nov 20 2006 *)

is(k) = sum(j = 1, 6, isprime(2*j*k+1)) == 6; \\ Jinyuan Wang, Aug 04 2019

Extended by Ray Chandler, Nov 20 2006

A102248 Numbers n such that n111 is prime.

Original entry on oeis.org

2, 4, 8, 10, 16, 22, 25, 26, 28, 35, 40, 44, 47, 50, 58, 65, 68, 70, 79, 80, 86, 92, 94, 95, 101, 109, 112, 113, 128, 131, 136, 140, 142, 143, 149, 152, 154, 169, 170, 179, 182, 184, 187, 196, 205, 208, 217, 218, 227, 235, 236, 260, 262, 263, 266, 278, 283, 284

Offset: 1

Parthasarathy Nambi, Feb 18 2005

			At n=2, n111 = 2111 (prime).
At n=50, n111 = 50111 (prime).
At n=95, n111 = 95111 (prime).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A024912.

Select[ Range[ 290], PrimeQ[ FromDigits[ Flatten[ IntegerDigits /@ { #, 1, 1, 1}]]] &] (* Robert G. Wilson v, Feb 21 2005 *)
Select[Range[300],PrimeQ[1000#+111]&] (* Harvey P. Dale, Jul 07 2019 *)

is(n)=isprime(1000*n+111) \\ Charles R Greathouse IV, May 22 2017

More terms from Robert G. Wilson v, Feb 21 2005

A101471 Numbers k such that the number k11 is prime.

Original entry on oeis.org

0, 2, 3, 8, 9, 15, 18, 20, 21, 23, 24, 27, 30, 35, 39, 41, 42, 50, 57, 60, 62, 63, 69, 72, 74, 80, 81, 83, 90, 93, 95, 98, 101, 102, 107, 113, 114, 120, 122, 125, 126, 129, 134, 137, 140, 144, 155, 161, 164, 168, 170, 179, 182, 183, 189, 192, 200, 204, 206, 210, 212

Offset: 1

Parthasarathy Nambi, Jan 30 2005, Feb 18 2005

k such that 100*k+11 is prime. - Robert Israel, Jul 30 2015

			If k=2,  then k11 =  211 (prime);
If k=50, then k11 = 5011 (prime);
If k=98, then k11 = 9811 (prime).

Robert Israel, Table of n, a(n) for n = 1..10000
Chris Caldwell, The first 1,000 primes.

Cf. A024912 (10n+1 is prime), A167442 (the actual primes 100n+11).

[n: n in [0..250] | IsPrime(Seqint(Intseq(11) cat Intseq(n)))]; // Vincenzo Librandi, Jul 31 2015

[n: n in [0..240] |IsPrime(100*n+11)]; // Vincenzo Librandi, Jul 31 2015

select(t -> isprime(100*t+11), [$0..1000]); # Robert Israel, Jul 30 2015

Select[ Range[0, 215], PrimeQ[ FromDigits[ Flatten[ IntegerDigits /@ { #, 1, 1}]]] &] (* Robert G. Wilson v, Feb 21 2005 *)
Select[Range[0,250],PrimeQ[100#+11]&] (* Harvey P. Dale, Oct 11 2018 *)

is(n)=isprime(100*n+11) \\ Charles R Greathouse IV, Feb 20 2017

Corrected and extended by Robert G. Wilson v, Feb 21 2005

cp's OEIS Frontend

A139368 a(n) = A024912(n) - A102342(n).

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A030430 Primes of the form 10*n+1.

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

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A081759 Numbers n such that 5n+6 is prime.

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A049511 Numbers k such that prime(k) == 1 (mod 10).

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Sage

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A023237 Primes p such that 10*p + 1 is also prime.

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Magma

Magma

Maple

Mathematica

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A124410 Numbers k such that 2k+1, 4k+1, 6k+1, 8k+1 and 10k+1 are primes.

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