A274397 -id:A274397 - 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

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

A003631 Primes congruent to 2 or 3 modulo 5.

Original entry on oeis.org

2, 3, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 103, 107, 113, 127, 137, 157, 163, 167, 173, 193, 197, 223, 227, 233, 257, 263, 277, 283, 293, 307, 313, 317, 337, 347, 353, 367, 373, 383, 397, 433, 443, 457, 463, 467, 487, 503, 523, 547, 557, 563, 577

Offset: 1

N. J. A. Sloane, Mira Bernstein

For n>1, sequence gives primes ending in 3 or 7. - Lekraj Beedassy, Oct 27 2003
Inert rational primes in Q(sqrt 5), or, p is not a square mod 5. [See e.g., Hasse, Legendre symbol (5|p) = -1, Hardy and Wright, Theorem 257 (2), p. 222, and Dodd Appendix B, pp. 128 - 150, primes p < 32771 with (p,0). - Wolfdieter Lang, Jun 16 2021]
Primes for which the period of the Fibonacci sequence mod p divides 2p+2.
Let F(n) be the n-th Fibonacci number for n=1,2,3,... (A000045). F(n) mod p (a prime) generates a periodic sequence. This sequence may be generated as follows: F(p-1)* F(p) mod p = p-1. E.g., p=7: F(6) * F(7) mod 7 = 8 * 13 mod 7 = 6 = p-1. - Louis Mello (Mellols(AT)aol.com), Feb 09 2001
These are also the primes p that divide Fibonacci(p+1). - Jud McCranie
Also primes p such that p divides F(2p+1)-1; such that p divides F(2p+3)-1; such that p divides F(3p+1)-1. - Benoit Cloitre, Sep 05 2003
Primes p such that the polynomial x^2-x-1 mod p has no zeros; i.e., x^2-x-1 is irreducible over the integers mod p. - T. D. Noe, May 02 2005
Primes p such that (1-x^5)/(1-x) is irreducible over GF(p). - Joerg Arndt, Aug 10 2011
Primes p such that p does not divide Sum_{i=1..p-1} Fibonacci(i)^2 = A001654(p-1). - Arkadiusz Wesolowski, Jul 23 2012
The prime 2 and primes p such that p^2 mod 10 = 9. - Richard R. Forberg, Aug 28 2013
Primes p such that 5 divides sigma(p^3), cf. A274397. - M. F. Hasler, Jul 10 2016

F. W. Dodd, Number Theory in the Quadratic Field with Golden Section Unit, Polygon Publishing House, Passaic, NJ 07055, 1983, Appendix B, pp. 128 - 150.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Chap. X, p. 150, Chap. XV, Theorem 257 (2), p. 222, Oxford University Press, Fifth edition.
H. Hasse, Number Theory, Springer-Verlag, NY, 1980, p. 498.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. N. Vorob'ev, Fibonacci Numbers, Pergamon Press, 1961.

T. D. Noe, Table of n, a(n) for n = 1..1000
Henri Darmon, Andrew Wiles's Marvelous Proof, Notices of the AMS (2017), Volume 64, Number 3 pp. 209-216. See p. 211.
Index to sequences related to decomposition of primes in quadratic fields

Primes in A047221.
Cf. A000040.
Cf. A000045, A001654.
Cf. A274397.

a003631 n = a003631_list !! (n-1)
a003631_list = filter ((== 1) . a010051') a047221_list
-- Reinhard Zumkeller, Nov 27 2012, Jul 19 2011

[ p: p in PrimesUpTo(1000) | p mod 5 in {2, 3} ]; // Vincenzo Librandi, Aug 07 2012

Select[ Prime[Range[106]], MemberQ[{2, 3}, Mod[#, 5]] &] (* Robert G. Wilson v, Sep 12 2011 *)
a[ n_] := If[ n < 1, 0, Module[{c = 0, m = 0}, While[ c < n, If[ PrimeQ[++m] && KroneckerSymbol[5, m] == -1, c++]]; m]]; (* Michael Somos, Nov 24 2018 *)

list(lim)=select(n->n%5==2||n%5==3,primes(primepi(lim))) \\ Charles R Greathouse IV, Jul 25 2011

{a(n) = if( n < 1, 0, my(c ,m); while( c < n, if( isprime(m++) && kronecker(5, m) == -1, c++)); m)}; /* Michael Somos, Aug 14 2012 */

a(n) ~ 2n log n. - Charles R Greathouse IV, Jun 19 2017

A274685 Odd numbers n such that sigma(n) is divisible by 5.

Original entry on oeis.org

19, 27, 29, 57, 59, 79, 87, 89, 95, 109, 133, 135, 139, 145, 149, 171, 177, 179, 189, 199, 203, 209, 229, 237, 239, 247, 261, 267, 269, 285, 295, 297, 319, 323, 327, 343, 349, 351, 359, 377, 379, 389, 395, 399, 409, 413, 417, 419, 435, 437, 439, 445, 447, 449, 459, 475, 479, 493, 499

Offset: 1

M. F. Hasler, Jul 02 2016

nonn

The subsequence of odd terms in A274397.
If n is in the sequence and gcd(n,m)=1 for some odd m, then n*m is also in the sequence. One might call "primitive" those terms which are not of this form, i.e., not a "coprime" multiple of an earlier term. The list of these primitive terms is (19, 27, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 343, 349, 359, 379, 389, 409, 419, 439, 449, 479, 499, ...). The primitive terms are the primes and powers of primes within the sequence. If a prime power p^k (k >= 1) is in the sequence, then p^(m(k+1)-1) is in the sequence for any m >= 1, since 1+p+...+p^(m(k+1)-1) = (1+p+...+p^k)(1+p^(k+1)+...+p^((m-1)*(k+1))). For example, with the prime p=19 we also have all odd powers 19^3, 19^5, ..., and with 27 = 3^3, we also have 27^5, 27^9, ... in the sequence.
On the other hand, for any prime p <> 5 there is an exponent k in {1, 3, 4} such that p^k is in this sequence (and therewith all higher powers of the form given above).
One may notice that there are many pairs of the form (30k-3, 30k-1), e.g., 27,29; 57,59; 87,89; 177,179; 237,239; 295,299; ... Indeed, it is likely that 30k-1 is prime and in this case, if 10k-1 is also prime, then sigma(30k-3) = 40k is divisible by 5 and sigma(30k-1) = 30k is also divisible by 5.

			Some values of a(2^k): a(2) = 27, a(4) = 57, a(8) = 89, a(16) = 171, a(32) = 297, a(64) = 545, a(128) = 1029, a(256) = 1937, a(512) = 3625, a(1024) = 6939, a(2048) = 13257, a(4096) = 25483, a(8192) = 49319, a(16384) = 95695, a(32768) = 185991, a(65536) = 362725, a(131072) = 708887, a(262144) = 1388367, a(524288) = 2722639, a(1048576) = 5346681, a(2097152) = 10514679, a(4194304) = 20698531, a(8388608) = 40790203.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000203 (sigma), A028983 (sigma even), A087943 (sigma = 3k), A248150 (sigma = 4k); A028982 (sigma is odd), A248151 (sigma is not divisible by 4); A272930(sigma(sigma(k)) = nk).

Select[Range[1, 500, 2], Divisible[DivisorSigma[1, #], 5] &] (* Michael De Vlieger, Jul 16 2016 *)

is_A274685(n)=sigma(n)%5==0&&bittest(n,0)

forstep(n=1,999,2,sigma(n)%5||print1(n","))

a(n) ~ 2n. - Charles R Greathouse IV, Jul 16 2016

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A030430 Primes of the form 10*n+1.

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A030433 Primes of form 10*k + 9.

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A003631 Primes congruent to 2 or 3 modulo 5.

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A274685 Odd numbers n such that sigma(n) is divisible by 5.

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