A007521 -id:A007521 - cp's OEIS frontend

A023230 Numbers k such that k and 8*k + 5 are both prime.

3, 7, 13, 19, 43, 67, 103, 109, 127, 139, 151, 181, 193, 199, 211, 223, 241, 277, 283, 349, 379, 397, 421, 433, 439, 463, 577, 601, 607, 613, 619, 727, 733, 787, 829, 853, 883, 967, 991, 1033, 1039, 1117, 1201, 1237, 1291, 1303, 1399, 1429, 1459, 1531, 1567, 1579

Offset: 1

David W. Wilson

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

Cf. A007521 (primes of form 8n+5), A105133 (numbers n such that 8n+5 is prime).

[ p: p in PrimesUpTo(1580) | IsPrime(8*p+5) ]; // Klaus Brockhaus, Dec 21 2008

lst={}; Do[p=Prime[n]; If[PrimeQ[8*p+5],AppendTo[lst,p]],{n,6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 16 2009 *)
Select[Prime [Range[0, 300]], PrimeQ[8# + 5] &] (* Vincenzo Librandi, Sep 25 2012 *)

select(p->isprime(8*p+5),primes(1000)) \\ Charles R Greathouse IV, Sep 25 2012

A105140 Numbers n such that 1024n+513 is prime.

Original entry on oeis.org

7, 10, 11, 17, 22, 25, 31, 35, 44, 50, 65, 74, 79, 82, 85, 94, 100, 109, 110, 112, 115, 116, 122, 130, 140, 149, 151, 154, 155, 157, 164, 166, 172, 179, 206, 211, 214, 215, 221, 227, 229, 232, 245, 254, 256, 259, 269, 271, 277, 280, 281, 292, 295, 296, 299, 316, 322, 332

Offset: 1

N. J. A. Sloane, based on correspondence from Marco Matosic, Apr 11 2005

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

Cf. A095278, A105133-A105139, A002145, A007521, A105126-A105132.

[n: n in [0..350] | IsPrime(1024*n + 513)]; // Vincenzo Librandi, Jan 08 2013

Select[Range[0, 500], PrimeQ[1024 # + 513]&] (* Vincenzo Librandi, Jan 08 2013 *)

is(n)=isprime(1024*n+513) \\ Charles R Greathouse IV, Feb 17 2017

A343113 Numbers having exactly 1 divisor of the form 8*k + 5.

Original entry on oeis.org

5, 10, 13, 15, 20, 21, 25, 26, 29, 30, 35, 37, 39, 40, 42, 50, 52, 53, 55, 58, 60, 61, 63, 69, 70, 74, 75, 77, 78, 80, 84, 87, 91, 93, 95, 100, 101, 104, 106, 109, 110, 111, 115, 116, 120, 122, 126, 133, 138, 140, 141, 143, 147, 148, 149, 150, 154, 155

Offset: 1

Jianing Song, Apr 05 2021

			52 is a term since among the divisors of 52 (namely 1, 2, 4, 13, 26 and 52), the only divisor congruent to 5 modulo 8 is 13.

Jianing Song, Table of n, a(n) for n = 1..10000

Numbers having m divisors of the form 8*k + i: A343107 (m=1, i=1), A343108 (m=0, i=3), A343109 (m=0, i=5), A343110 (m=0, i=7), A343111 (m=2, i=1), A343112 (m=1, i=3), this sequence (m=1, i=5), A141164 (m=1, i=7).
Indices of 1 in A188171.
A007521 is a subsequence.

res(n,a,b) = sumdiv(n, d, (d%a) == b)
isA343113(n) = (res(n,8,5) == 1)

A175461 Semiprimes of form 8n+5.

Original entry on oeis.org

21, 69, 77, 85, 93, 133, 141, 205, 213, 221, 237, 253, 301, 309, 341, 365, 381, 413, 437, 445, 453, 469, 485, 493, 501, 517, 533, 565, 573, 581, 589, 597, 629, 669, 685, 717, 749, 781, 789, 813, 869, 893, 901, 917, 933, 949, 965, 973, 989, 1037, 1077, 1101

Offset: 1

Zak Seidov, May 23 2010

nonn

There are no squares of form 8n+5.

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

Cf. A001358 Semiprimes: products of two primes, A004770 Numbers of form 8n+5, A007521 Primes of form 8n+5, A175462 Number of divisors of integers of form 8n+5, A175463 Numbers n such that 8*n+5 is semiprime.

Select[8*Range[200]+5,PrimeOmega[#]==2&] (* Harvey P. Dale, Apr 29 2015 *)

lista(nn) = {vec = vector(nn, i, 8*i+5); smp = select(i->(bigomega(i) == 2), vec); print(smp);} \\ Michel Marcus, Oct 15 2013

A263011 Numbers D == 1 (mod 8), not a square, and if composite without prime factors 3 or 5 (mod 8).

Original entry on oeis.org

17, 41, 73, 89, 97, 113, 137, 161, 193, 217, 233, 241, 257, 281, 313, 329, 337, 353, 401, 409, 433, 449, 457, 497, 521, 553, 569, 577, 593, 601, 617, 641, 673, 697, 713, 721, 761, 769, 809, 833, 857, 881, 889, 929, 937, 953, 977, 1009, 1033, 1049, 1057, 1081, 1097, 1129, 1153, 1169, 1193, 1201, 1217

Offset: 1

Wolfdieter Lang, Nov 17 2015

These numbers are the odd D candidates for the (generalized) Pell equation x^2 - D*y^2 = +8 which could have proper solutions (x, y) with x and y both odd (and gcd(x, y) = 1).
Proof: Put x =2*X + 1, y = 2*Y + 1; then 8*(T(X) - D*T(Y)) = 8 - 1 + D = 7 + D, with the triangular numbers T = A000217. Hence, D == -7 (mod 8) == +1 (mod 8). Only nonsquare numbers D are considered for the Pell equation (square D leads to a factorization with only one solution: D = 1, (x, y) = (3, 1)). Consider a prime factor p == 3 or 5 (mod 8) (A007520 or A007521) of D. Then x^2 == 8 (mod p). Because the Legendre symbol (8/p) = (2*2^2/p) = (2/p) ==  (-1)^(p^2-1)/8 (see, e.g., Nagell, eq. (3), p. 138) this becomes -1 for these primes p, and therefore a candidate for D cannot have any prime factors 3 or 5 (mod 8).
However, not all of these candidates admit solutions. For the exceptions see A264348.
The remaining Ds (that admit solutions) are given in A263012.

T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A263012, A264348.

Select[8 Range@ 154 + 1, Or[PrimeQ@ #, CompositeQ@ # && AllTrue[Union@ Mod[First /@ FactorInteger@ #, 8], ! MemberQ[{3, 5}, #] &]] && ! IntegerQ@ Sqrt@ # &] (* Michael De Vlieger, Dec 11 2015, Version 10 *)

A045323 Primes congruent to {1, 2, 3, 7} (mod 8).

Original entry on oeis.org

2, 3, 7, 11, 17, 19, 23, 31, 41, 43, 47, 59, 67, 71, 73, 79, 83, 89, 97, 103, 107, 113, 127, 131, 137, 139, 151, 163, 167, 179, 191, 193, 199, 211, 223, 227, 233, 239, 241, 251, 257, 263, 271, 281, 283, 307, 311

Offset: 1

N. J. A. Sloane

Numbers x such that x^4 = 4 has a solution mod p.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

Cf. A000040, A010051.

Subsequence of A004776; see also A007521.

a045323 n = a045323_list !! (n-1)
a045323_list = filter ((== 1). a010051) a004776_list
-- Reinhard Zumkeller, Aug 17 2012

[ p: p in PrimesUpTo(400) | p mod 8 in {1, 2, 3, 7} ]; // Vincenzo Librandi, Aug 07 2012

Select[Prime[Range[300]],MemberQ[{1,2,3,7},Mod[#,8]]&] (* Vincenzo Librandi, Aug 07 2012 *)

A105131 Primes of the form 512n+257.

Original entry on oeis.org

257, 769, 3329, 7937, 9473, 14081, 14593, 22273, 23297, 26881, 30977, 31489, 36097, 37633, 40193, 41729, 43777, 46337, 49409, 49921, 57089, 57601, 60161, 70913, 75521, 77569, 78593, 84737, 88321, 91393, 96001, 98561, 100609, 103681, 106753, 107777

Offset: 1

N. J. A. Sloane, based on correspondence from Marco Matosic, Apr 11 2005

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

Cf. A002145, A007521, A105126-A105132.

[ a: n in [0..210] | IsPrime(a) where a is 512*n+257 ]; // Vincenzo Librandi, Jul 19 2012

Select[Table[512*n+257,{n,0,800}],PrimeQ] (* Vincenzo Librandi, Jul 19 2012 *)

A343104 Smallest number having exactly n divisors of the form 8*k + 1.

Original entry on oeis.org

1, 9, 81, 153, 891, 1377, 8019, 3825, 11025, 15147, 88209, 31977, 354375, 99225, 121275, 95931, 7144929, 187425, 893025, 287793, 1403325, 1499553, 1715175, 675675, 1091475, 6024375, 1576575, 1686825, 72335025, 2027025, 2264802453041139, 2297295, 11609325, 121463793, 9823275

Offset: 1

Jianing Song, Apr 05 2021

nonn

Smallest index of n in A188169.
a(n) exists for all n, since 3^(2n-2) has exactly n divisors of the form 8*k + 1, namely 3^0, 3^2, ..., 3^(2n-2). This actually gives an upper bound for a(n).
From David A. Corneth, Apr 05 2021: (Start)
All terms are odd since if a term is even then the odd part has the same number of such divisors.
No a(2*k + 1) is divisible by a prime congruent to 1 (mod 8).
If for some k, A188169(k) > m then A188169(k*t) > m for all t > 0. This can be used to trim searches when looking for some a(m).
If gcd(k, m) = 1 then A188169(k) * A188169(m) <= A188169(k*m) (End)

			a(4) = 153 since it is the smallest number with exactly 4 divisors congruent to 1 modulo 8, namely 1, 9, 17 and 153.

Smallest number having exactly n divisors of the form 8*k + i: this sequence (i=1), A343105 (i=3), A343106 (i=5), A188226 (i=7).
Cf. A188169.
Cf. A007519, A007520, A007521, A007522.

res(n,a,b) = sumdiv(n, d, (d%a) == b)
a(n) = if(n>0, for(k=1, oo, if(res(k,8,1)==n, return(k))))

a(2n-1) <= 3^(2n-2) * 11, since 3^(2n-2) * 11 has exactly 2n-1 divisors congruent to 1 modulo 8: 3^0, 3^2, ..., 3^(2n-2), 3^1 * 11, 3^3 * 11, ..., 3^(2n-3) * 11.

a(2n) <= 3^(n-1) * 187, since 3^(n-1) * 187 has exactly 2n divisors congruent to 1 modulo 8: 3^0, 3^2, ..., 3^b, 3^0 * 17, 3^2 * 17, ..., 3^b * 17, 3^1 * 11, 3^3 * 11, ..., 3^a * 11, 3^1 * 187, 3^3 * 187, ... 3^a * 187, where a is the largest odd number <= n-1 and b is the largest even number <= n-1.

More terms from David A. Corneth, Apr 06 2021

A105127 Primes of the form 32n+17.

Original entry on oeis.org

17, 113, 241, 337, 401, 433, 593, 881, 977, 1009, 1201, 1297, 1361, 1489, 1553, 1777, 1873, 2129, 2161, 2417, 2609, 2801, 2833, 2897, 3089, 3121, 3217, 3313, 3697, 3761, 3793, 3889, 4049, 4177, 4241, 4273, 4337, 4561, 4657, 4721, 4817, 5009, 5233, 5297

Offset: 1

N. J. A. Sloane, based on correspondence from Marco Matosic, Apr 11 2005

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

Cf. A002145, A007521, A105126-A105132.

[ a: n in [0..200] | IsPrime(a) where a is 32*n+17 ]; // Vincenzo Librandi, Jul 19 2012

Select[Range[17,20000,32],PrimeQ] (* Vladimir Joseph Stephan Orlovsky, May 02 2011 *)
Select[Table[32*n+17,{n,0,700}],PrimeQ] (* Vincenzo Librandi, Jul 19 2012 *)

A105129 Primes of the form 128n+65.

Original entry on oeis.org

193, 449, 577, 1217, 1601, 2113, 2753, 3137, 4289, 4673, 4801, 5441, 5569, 5953, 6337, 6977, 7489, 7873, 8513, 8641, 9281, 10177, 10433, 11329, 11969, 12097, 13121, 13249, 13633, 14401, 14657, 15809, 15937, 16193, 17729, 19009, 19777, 20161, 20929, 21313

Offset: 1

N. J. A. Sloane, based on correspondence from Marco Matosic, Apr 11 2005

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

Cf. A002145, A007521, A105126-A105132.

[ a: n in [0..200] | IsPrime(a) where a is 128*n+65 ]; // Vincenzo Librandi, Jul 19 2012

Select[128*Range[0,200]+65,PrimeQ] (* Harvey P. Dale, Apr 17 2012 *)
Select[Table[128*n+65,{n,0,700}],PrimeQ] (* Vincenzo Librandi, Jul 19 2012 *)

cp's OEIS Frontend

A023230 Numbers k such that k and 8*k + 5 are both prime.

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A105140 Numbers n such that 1024n+513 is prime.

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A343113 Numbers having exactly 1 divisor of the form 8*k + 5.

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A175461 Semiprimes of form 8n+5.

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A263011 Numbers D == 1 (mod 8), not a square, and if composite without prime factors 3 or 5 (mod 8).

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A045323 Primes congruent to {1, 2, 3, 7} (mod 8).

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A105131 Primes of the form 512n+257.

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Magma

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A343104 Smallest number having exactly n divisors of the form 8*k + 1.

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