A007520 -id:A007520 - cp's OEIS frontend

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

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 *)

A023229 Primes p such that 8*p + 3 is also prime.

Original entry on oeis.org

2, 5, 7, 13, 17, 31, 41, 43, 47, 61, 71, 73, 101, 103, 107, 113, 127, 131, 157, 163, 181, 191, 197, 223, 233, 241, 251, 281, 283, 293, 307, 317, 337, 367, 383, 421, 433, 443, 457, 461, 467, 491, 503, 563, 631, 643, 647, 653, 673, 677, 691, 733, 751, 761, 787, 797, 811

Offset: 1

David W. Wilson

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

Cf. A005124, A007520.

[n: n in PrimesUpTo(1000) | IsPrime(8*n+3)]; // Vincenzo Librandi, Nov 20 2010

a := proc (n) if isprime(n) = true and isprime(8*n+3) = true then n else end if end proc: seq(a(n), n = 1 .. 900); # Emeric Deutsch, Dec 26 2008

Select[Prime@Range@500, PrimeQ[8 # + 3] &] (* Vincenzo Librandi, May 19 2014 *)

A045339 Primes congruent to {2, 3} mod 8.

Original entry on oeis.org

2, 3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251, 283, 307, 331, 347, 379, 419, 443, 467, 491, 499, 523, 547, 563, 571, 587, 619, 643, 659, 683, 691, 739, 787, 811, 827, 859, 883, 907, 947

Offset: 1

N. J. A. Sloane

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

Cf. A007520.

[ p: p in PrimesUpTo(1200) | p mod 8 in {2, 3} ]; // Vincenzo Librandi, Aug 07 2012

Select[Prime[Range[200]],MemberQ[{2,3},Mod[#,8]]&] (* Harvey P. Dale, Oct 31 2011 *)

{2} UNION A007520. - R. J. Mathar, Dec 04 2011

A332446 Numbers k for which A087808(sigma(k)) is equal to A087808(2*k).

Original entry on oeis.org

3, 6, 11, 19, 28, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 216, 227, 251, 267, 283, 286, 307, 331, 347, 379, 419, 443, 467, 491, 496, 499, 523, 547, 563, 571, 587, 598, 619, 643, 659, 683, 691, 726, 739, 787, 811, 827, 859, 883, 907, 947, 971, 1019, 1051, 1091, 1123, 1163, 1171, 1187, 1259, 1283, 1291, 1307, 1427, 1451

Offset: 1

Antti Karttunen, Feb 14 2020

nonn

Conjecture: includes all terms of A007520. - Bill McEachen, Dec 10 2023

Cf. A000203, A007520, A087808.
Subsequences: A000396, A332445.
Cf. A331751, A331752, A332208 for similar sequences.

A087808(n) = if(n<1, 0, if(n%2==0, 2*A087808(n/2), A087808((n-1)/2)+1));
isA332446(n) = (A087808(2*n)==A087808(sigma(n)));

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

A178985 Primes of the form 3^k mod 2^k, in the order in which they are found.

Original entry on oeis.org

3, 19, 11, 227, 1019, 269201, 186023729, 457933343698297657, 2267602862220213494836920572800947269169358383491, 3510117420185552058703020362961660520827436011216742688744177

Offset: 1

Juri-Stepan Gerasimov, Jan 03 2011

nonn

Can it be shown that this is always an increasing sequence?
{a(n)} is an increasing sequence because {a(n)} is a subsequence of the integer sequence {b(n)} = (fractional part of (3/2)^n without the decimal point)/5^n = A204544(n) / 5^n = prime terms of A002380. - Michel Lagneau, Jan 25 2012
Corresponding n: 3, 5, 7, 9, 11, 20, 28, 62, 161, 204, 471, 505, 881, 1810, 1812, 2506, 3321, ... - Eric Chen, Jun 13 2018

Eric Chen, Table of n, a(n) for n = 1..17

Cf. A002380, A007520, A204544.

f[n_] := PowerMod[3, n, 2^n]; Select[f@ Range@ 300, PrimeQ]

A201914 Least prime p such that p+1 is divisible by 2^n and not by 2^(n+1).

Original entry on oeis.org

2, 5, 3, 7, 47, 31, 191, 127, 1279, 3583, 5119, 6143, 20479, 8191, 81919, 294911, 1114111, 131071, 786431, 524287, 17825791, 14680063, 138412031, 109051903, 654311423, 1912602623, 738197503, 2818572287, 7247757311, 3758096383, 228707008511, 2147483647

Offset: 0

T. D. Noe, Dec 27 2011

nonn

See A126717 for the least k such that k*2^n-1 is prime.

For every n >= 1 there are infinitely many prime numbers p such that p + 1 is divisible by 2^n and not by 2^(n + 1). - Marius A. Burtea, Mar 10 2020

Laurențiu Panaitopol, Alexandru Gica, Arithmetic problems and number theory, Ed. Gil, Zalău, (2006), ch. 13, p. 78, pr. 5 (in Romanian).

Donovan Johnson, Table of n, a(n) for n = 0..1000

Cf. A008864 (primes + 1), A057775 (p-1 case), A126717.

For n>0, sequence is first term of A002144, A007520, A141194, A142041, A142939, ...

a:=[]; for n in [0..31] do k:=1; while not IsPrime(k*2^n-1) do k:=k+2; end while; Append(~a,k*2^n-1); end for; a; // Marius A. Burtea, Mar 10 2020

Table[k = 1; While[p = k*2^n - 1; ! PrimeQ[p], k = k + 2]; p, {n, 0, 40}]

A243177 Numbers of the form 3x^2+2xy+3y^2.

Original entry on oeis.org

0, 3, 4, 8, 11, 12, 16, 19, 24, 27, 32, 36, 43, 44, 48, 51, 59, 64, 67, 68, 72, 75, 76, 83, 88, 96, 99, 100, 107, 108, 123, 128, 131, 132, 136, 139, 144, 147, 152, 163, 164, 171, 172, 176, 179, 187, 192, 196, 200, 204, 211, 216, 219, 227, 228, 236, 243, 251, 256, 264, 267, 268, 272, 275, 283, 288, 291, 292, 300, 304, 307, 323, 324, 328, 331, 332, 339, 344, 347

Offset: 1

N. J. A. Sloane, Jun 02 2014

nonn

Discriminant -32.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Naoki Uchida, Integers of the Form ax^2 + bxy + cy^2 (2020 preprint)

Primes: A007520.

ofTheFormQ[n_] := Reduce[n == 3*x^2 + 2*x*y + 3*y^2, {x, y}, Integers] =!= False; Select[Range[0, 400], ofTheFormQ] (* Jean-François Alcover, Jun 04 2014 *)

is(n)=if(n==0, return(1)); my(h=valuation(n,2),f=factor(n>>h),s); if(h==1, return(0)); for(i=1,#f~, if(f[i,1]%8==3, s+=f[i,2], f[i,1]%8>3 && f[i,2]%2, return(0))); h>1 || s%2 \\ Charles R Greathouse IV, Feb 10 2020

A294912 Numbers n such that 2^(n-1), (2n-1)(2^((n-1)/2)), (4ceiling((3/4)n)-2), and (2^((n+1)/2) + floor((1/4)n)2^(((n+1)/2)+1)) are all congruent to 1 (mod n).

Original entry on oeis.org

3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251, 283, 307, 331, 347, 379, 419, 443, 467, 491, 499, 523, 547, 563, 571, 587, 619, 643, 659, 683, 691, 739, 787, 811, 827, 859, 883, 907, 947, 971, 1019, 1051, 1091, 1123, 1163, 1171, 1187, 1259

Offset: 1

Jonas Kaiser, Nov 10 2017

nonn

It appears that A007520 is a subsequence.
The first composite term is a(9969) = 476971 = 11*131*331. - Alois P. Heinz, Nov 10 2017
From Hilko Koning, Dec 03 2019: (Start)
The next composite terms < 1999979 are
a(17428) = 877099  = 307*2857
a(25090) = 1302451 = 571*2281
a(25518) = 1325843 = 499*2657
a(26785) = 1397419 = 67*20857
a(27549) = 1441091 = 347*4153
a(28715) = 1507963 = 971*1553
a(29117) = 1530787 = 619*2473
a(35635) = 1907851 = 11*251*691
(End)
From Hilko Koning, Dec 05 2019: (Start)
The next composite terms < 24999971 are
a(37344) = 2004403 = 307*6529
a(55773) = 3090091 = 1163*2657
a(56189) = 3116107 = 883*3529
a(91332) = 5256091 = 811*6481
a(102027) = 5919187 = 1777*3331
a(133230) = 7883731 = 811*9721
a(156407) = 9371251 = 1531*6121
a(182911) = 11081459 = 227*48817
a(189922) = 11541307 = 1699*6793
a(201043) = 12263131 = 811*15121
a(213203) = 13057787 = 467*27961
a(217484) = 13338371 = 3163*4217
a(257526) = 15976747 = 3739*4273
a(274961) = 17134043 = 1097*15619
a(299096) = 18740971 = 1531*12241
a(308928) = 19404139 = 2011*9649
a(321676) = 20261251 = 2251*9001
a(341902) = 21623659 = 1163*18593
a(348622) = 22075579 = 163*135433
a(380162) = 24214051 = 281*86171
The composite terms < 25*10^6 match the terms of A244628.
(End)
It appears that composites of the form 2k+1 such that 3*(2k+1) divides 2^k+1 are the composite terms of this sequence. - Hilko Koning, Dec 09 2019

Jonas Kaiser, On the relationship between the Collatz conjecture and Mersenne prime numbers, arXiv:1608.00862 [math.GM], 2016.

Cf. A244626, A294717, A293394, A070179, A007520.

okQ[n_] := AllTrue[{2^(n-1), (2*n-1)*(2^((n-1)/2)), (4*Ceiling@((3/4)*n) - 2), (2^((n+1)/2) + Floor@(n/4)*2^(((n+1)/2)+1))}, Mod[#, n] == 1&];
Select[Range[1300], okQ] (* Jean-François Alcover, Feb 18 2019 *)

isok(n) = (n%2) && lift((Mod(2, n)^(n-1))==1)&&lift((Mod((2*n-1), n)*Mod(2, n)^((n-1)/2)) == 1)&&lift((Mod(((4*ceil((3/4)*n)-2)), n) )== 1)&&lift((Mod(2, n)^((n+1)/2) +Mod(floor((1/4)*n),n)*Mod(2, n)^(((n+1)/2)+1 ))== 1)

More terms from Alois P. Heinz, Nov 10 2017

A336792 Values of odd prime numbers, D, for incrementally largest values of minimal positive y satisfying the equation x^2 - D*y^2 = -2.

Original entry on oeis.org

3, 19, 43, 67, 139, 211, 331, 379, 571, 739, 859, 1051, 1291, 1531, 1579, 1699, 2011, 2731, 3019, 3259, 3691, 3931, 5419, 5659, 5779, 6211, 6379, 6451, 8779, 9619, 10651, 16699, 17851, 18379, 21739, 25939, 32971, 42331, 42571, 44851, 50131, 53299, 55819, 56611, 60811, 61051, 73459, 76651, 90619, 90931

Offset: 1

Christine Patterson, Oct 14 2020

nonn

For the corresponding y values see A336793.

For solutions of this Diophantine equation it is sufficient to consider the odd primes p(n) := A007520(n), for n >= 1, the primes 3 (mod 8). Also prime 2 has the fundamental solution (x, y) = (0, 1). If there is a solution for p(n) then there is only one infinite family of solutions because there is only one representative parallel primitive binary quadratic form for Discriminant Disc = 4*p(n) and representation k = -2. Only proper solutions can occur. The conjecture is that each p(n) leads to solutions. For the fundamental solutions (with prime 2) see A339881 and A339882. - Wolfdieter Lang, Dec 22 2020

			For D=3, the least positive y for which x^2-D*y^2=-2 has a solution is 1. The next prime, D, for which x^2-D*y^2=-2 has a solution is 11, but the smallest positive y in this case is also 1, which is equal to the previous record y. So 11 is not a term.
The next prime, D, after 11 for which x^2-D*y^2=-2 has a solution is 19 and the least positive y for which it has a solution is y=3, which is larger than 1, so it is a new record y value. So 19 is a term of this sequence and 3 is a term of A336793.

Christine Patterson, Sage Program

Cf. A033316 (analogous for x^2-D*y^2=1), A336790 (similar sequence for x's), A336793.

cp's OEIS Frontend

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

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A023229 Primes p such that 8*p + 3 is also prime.

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Magma

Maple

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A045339 Primes congruent to {2, 3} mod 8.

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Magma

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A332446 Numbers k for which A087808(sigma(k)) is equal to A087808(2*k).

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PARI

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

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Extensions

A178985 Primes of the form 3^k mod 2^k, in the order in which they are found.

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Mathematica

A201914 Least prime p such that p+1 is divisible by 2^n and not by 2^(n+1).

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Magma

Mathematica

A243177 Numbers of the form 3x^2+2xy+3y^2.

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Author

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Programs

Mathematica

A294912 Numbers n such that 2^(n-1), (2n-1)(2^((n-1)/2)), (4ceiling((3/4)n)-2), and (2^((n+1)/2) + floor((1/4)n)2^(((n+1)/2)+1)) are all congruent to 1 (mod n).