| Displaying 1-4 of 4 results found.
page
1
Primes of the form 10*n+1.
+10
77
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
COMMENTS
Also primes of form 5*n+1 or equivalently 5*n+6.
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
MATHEMATICA
Select[Prime@Range[210], Mod[ #, 10] == 1 &] (* Ray Chandler, Dec 06 2006 *) Select[Range[11, 1291, 10], PrimeQ] (* Zak Seidov, Aug 14 2011*)
PROG
(Haskell)
a030430 n = a030430_list !! (n-1)
a030430_list = filter ((== 1) . a010051) a017281_list
(PARI) lista(nn) = forprime(p=11, nn, if(p%10==1, print1(p, ", "))) \\ Iain Fox, Dec 30 2017
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
COMMENTS
Also primes of form 5*k + 4.
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 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)
MAPLE
select(isprime, [seq(10*n+9, n=1..500)]); # Muniru A Asiru, Apr 27 2018
MATHEMATICA
Select[Prime@Range[210], Mod[ #, 10] == 9 &] (* Ray Chandler, Nov 07 2006 *) Prime@Flatten@Position[Length@FactorList[((1+d)^5-1)/d, Modulus->#]&/@Prime@Range@200, 3] (* Federico Provvedi, Apr 04 2018 *)
PROG
(PARI) for(n=1, 1e3, if(isprime(p=10*n+9), print1(p, ", "))); \\ Altug Alkan, Apr 19 2018 (GAP) Filtered(List([1..500], n->10*n+9), IsPrime); # Muniru A Asiru, Apr 27 2018
Primes congruent to 2 or 3 modulo 5.
(Formerly M0832)
+10
44
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
COMMENTS
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 McCranieAlso 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
REFERENCES
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.
MATHEMATICA
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 *)
PROG
(Haskell)
a003631 n = a003631_list !! (n-1)
a003631_list = filter ((== 1) . a010051') a047221_list
(PARI) {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 */ (Magma) [ p: p in PrimesUpTo(1000) | p mod 5 in {2, 3} ]; // Vincenzo Librandi, Aug 07 2012
Odd numbers n such that sigma(n) is divisible by 5.
+10
2
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
COMMENTS
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.
EXAMPLE
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.
MATHEMATICA
Select[Range[1, 500, 2], Divisible[DivisorSigma[1, #], 5] &] (* Michael De Vlieger, Jul 16 2016 *)
PROG
(PARI) is_ A274685(n)=sigma(n)%5==0&&bittest(n, 0) (PARI) forstep(n=1, 999, 2, sigma(n)%5||print1(n", "))
Search completed in 0.007 seconds
|