A081759 -id:A081759 - cp's OEIS frontend

A030430 Primes of the form 10*n+1.

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

A024912 Numbers k such that 10*k + 1 is prime.

Original entry on oeis.org

1, 3, 4, 6, 7, 10, 13, 15, 18, 19, 21, 24, 25, 27, 28, 31, 33, 40, 42, 43, 46, 49, 52, 54, 57, 60, 63, 64, 66, 69, 70, 75, 76, 81, 82, 88, 91, 94, 97, 99, 102, 103, 105, 106, 109, 115, 117, 118, 120, 123, 129, 130, 132, 136, 138, 145, 147, 148, 151, 153, 157, 160, 162

Offset: 1

Clark Kimberling

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A030430, A049511, A081759.

[n: n in [1..1000] |IsPrime(10*n+1)]; // Vincenzo Librandi, Nov 19 2010

Select[Range[170], PrimeQ[10# + 1] &] (* Ray Chandler, Dec 06 2006 *)

is(n)=isprime(10*n+1) \\ Charles R Greathouse IV, Jul 02 2013

a(n) = (A081759(n)+1)/2.

A023219 Primes p such that 5p+6 is a prime.

Original entry on oeis.org

5, 7, 11, 13, 19, 29, 37, 41, 47, 53, 61, 79, 83, 97, 103, 107, 113, 127, 131, 137, 139, 149, 151, 163, 181, 193, 197, 211, 229, 233, 239, 257, 263, 271, 293, 313, 347, 359, 373, 379, 389, 401, 421, 431, 443, 449, 461, 467, 487, 503, 509, 523, 541, 547, 557, 569, 571, 593

Offset: 1

David W. Wilson

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
J. Chavoya, A. Lucero, S. Reynolds, On p-adic valuations of the generalized Fibonacci sequences, in MSRI-UP Research Reports, 2014, pp. 91-109.

Subset of A081759.

Cf. A090161, A088555, A088561.

[p: p in PrimesUpTo(200)| IsPrime(5*p+6)]; // Vincenzo Librandi, Nov 20 2010

Select[Prime[Range[200]],PrimeQ[5#+6]&] (* Harvey P. Dale, Jul 27 2011 *)

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

A111225 Numbers n such that 5*n + 8 is prime.

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 19, 21, 31, 33, 37, 43, 45, 51, 55, 57, 61, 69, 73, 75, 85, 87, 91, 99, 103, 111, 117, 121, 127, 129, 133, 135, 145, 147, 153, 163, 169, 171, 175, 189, 195, 201, 205, 211, 217, 219, 223, 229, 231, 237, 241, 243, 255, 259, 273, 283, 285, 289

Offset: 1

Parthasarathy Nambi, Oct 26 2005

nonn

			If n=103 then 5*n + 8 = 523 (prime).

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

Cf. A024894, A087505, A024897, A081759.

[n: n in [0..100000] |IsPrime(5*n+8)]; // Vincenzo Librandi, Sep 03 2010

Select[Range[300],PrimeQ[5#+8]&] (* Harvey P. Dale, Oct 31 2011 *)

is(n)=isprime(5*n+8) \\ Charles R Greathouse IV, Feb 17 2017

A023254 Primes that remain prime through 2 iterations of the function f(x) = 5x + 6.

Original entry on oeis.org

7, 11, 29, 41, 53, 79, 83, 107, 113, 137, 139, 149, 163, 181, 193, 233, 239, 359, 379, 401, 449, 487, 503, 547, 601, 643, 653, 809, 839, 877, 883, 911, 1009, 1087, 1093, 1103, 1117, 1171, 1259, 1289, 1303, 1367, 1373, 1481, 1511, 1579, 1637, 1747, 1787, 1801

Offset: 1

David W. Wilson

nonn

Primes p such that 5*p+6 and 25*p+36 are also primes. - Vincenzo Librandi, Aug 04 2010

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

Subsequence of A081759.

[n: n in [0..100000] | IsPrime(n) and IsPrime(5*n+6) and IsPrime(25*n+36)] // Vincenzo Librandi, Aug 04 2010

A023285 Primes that remain prime through 3 iterations of function f(x) = 5x + 6.

Original entry on oeis.org

7, 79, 181, 233, 359, 401, 449, 1009, 1093, 1259, 1303, 1373, 1511, 1931, 2011, 2339, 2477, 3019, 3691, 4349, 4409, 5417, 5879, 6301, 6553, 6637, 7079, 8329, 9127, 9137, 10303, 10499, 11579, 12391, 13259, 14251, 15101, 15107, 15217, 15329, 15527, 15679

Offset: 1

David W. Wilson

nonn

Primes p such that 5*p+6, 25*p+36 and 125*p+186 are also primes. - Vincenzo Librandi, Aug 04 2010

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

Subsequence of A023219, A023254, and of A081759.

[n: n in [1..150000] | IsPrime(n) and IsPrime(5*n+6) and IsPrime(25*n+36) and IsPrime(125*n+186)] // Vincenzo Librandi, Aug 04 2010

p3Q[n_]:=And@@PrimeQ/@NestList[5#+6&,n ,3]; Select[Prime[Range[2000]],p3Q] (* Harvey P. Dale, Feb 20 2011 *)

A111224 Numbers n such that 5*n + 7 is prime.

Original entry on oeis.org

0, 2, 6, 8, 12, 18, 20, 24, 26, 30, 32, 38, 44, 50, 54, 60, 62, 66, 68, 72, 78, 90, 92, 96, 108, 110, 114, 116, 120, 122, 128, 134, 144, 150, 156, 158, 164, 170, 174, 176, 180, 186, 188, 192, 194, 198, 216, 218, 222, 236, 242, 246, 254, 258, 260, 264, 272, 284

Offset: 1

Parthasarathy Nambi, Oct 26 2005

			If n=108 then 5*n + 7 = 547 (prime).

Cf. A024894, A087505, A024897, A081759.

[n: n in [0..300] | IsPrime(5*n+7)]; // Vincenzo Librandi, Sep 03 2010

Select[Range[0,300],PrimeQ[5#+7]&] (* Harvey P. Dale, Jun 24 2018 *)

is(n)=isprime(5*n+7) \\ Charles R Greathouse IV, Jun 12 2017

A023315 Primes that remain prime through 4 iterations of function f(x) = 5*x + 6.

Original entry on oeis.org

79, 401, 1259, 2477, 3019, 4409, 10303, 15679, 20509, 24499, 34127, 43987, 44389, 53101, 66359, 71287, 74857, 81097, 85903, 90803, 93053, 102811, 103231, 104999, 112601, 125453, 132533, 144731, 156347, 157793, 160817, 161839, 163981, 170641

Offset: 1

David W. Wilson

nonn

Primes p such that 5*p+6, 25*p+36, 125*p+186 and 625*p+936 are also primes. - Vincenzo Librandi, Aug 04 2010

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

Subsequence of A023219, A023254, A023285, and A081759.

[n: n in [1..1000000] | IsPrime(n) and IsPrime(5*n+6) and IsPrime(25*n+36) and IsPrime(125*n+186) and IsPrime(625*n+936)]; // Vincenzo Librandi, Aug 04 2010

a(n) == 9 or 13 (mod 14). - John Cerkan, Oct 07 2016

A023343 Primes that remain prime through 5 iterations of function f(x) = 5x + 6.

Original entry on oeis.org

79, 34127, 345431, 549089, 669937, 703663, 948593, 978749, 999007, 1251329, 1255333, 1279133, 1500277, 1517413, 1525421, 1642769, 1670629, 1688101, 1727161, 1770127, 2152159, 2161343, 2328517, 2622167, 2745451, 2786681, 2837557, 3281777

Offset: 1

David W. Wilson

nonn

Primes p such that 5*p+6, 25*p+36, 125*p+186, 625*p+936 and 3125*p+4686 are also primes. - Vincenzo Librandi, Aug 05 2010

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

Subsequence of A023219, A023254, A023285, A023315, and A081759.

[n: n in [1..10000000] | IsPrime(n) and IsPrime(5*n+6) and IsPrime(25*n+36) and IsPrime(125*n+186) and IsPrime(625*n+936) and IsPrime(3125*n+4686)] // Vincenzo Librandi, Aug 05 2010

Select[Prime[Range[250000]],And@@PrimeQ[Rest[NestList[5#+6&,#,5]]]&] (* Harvey P. Dale, Jan 03 2014 *)

a(n) == 9 (mod 14). - John Cerkan, Oct 20 2016

cp's OEIS Frontend

A030430 Primes of the form 10*n+1.

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A024912 Numbers k such that 10*k + 1 is prime.

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A023219 Primes p such that 5p+6 is a prime.

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

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A111225 Numbers n such that 5*n + 8 is prime.

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A023254 Primes that remain prime through 2 iterations of the function f(x) = 5x + 6.

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Magma

A023285 Primes that remain prime through 3 iterations of function f(x) = 5x + 6.

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Magma

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A111224 Numbers n such that 5*n + 7 is prime.

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