A002313 -id:A002313 - cp's OEIS frontend

A145297 Smallest k such that k^2+1 is divisible by A002144(n)^4.

182, 239, 27493, 34522, 800982, 1251967, 623098, 6304056, 6459524, 20099637, 22709274, 35764191, 40317977, 54397650, 166206108, 187800003, 165728858, 152475014, 282599844, 312923750, 154613663, 485200742, 912190662, 548850444

Offset: 1

Klaus Brockhaus, Oct 11 2008

nonn

			a(1) = 182 since A002144(1) = 5, 182^2+1 = 33125 = 5^4*53 and for no k < 182 does 5^4 divide k^2+1.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A002144 (primes of form 4n+1), A002313 (-1 is a square mod p), A059321, A145296, A145298, A145299.

{e=4; forprime(p=2, 250, if(p%4==1, q=p^e; m=q; while(!ispower(m-1,2,&n), m=m+q); print1(n, ",")))}

from itertools import islice
from sympy import nextprime, sqrt_mod_iter
def A145297_gen(): # generator of terms
    p = 1
    while (p:=nextprime(p)):
        if p&3==1:
            yield min(sqrt_mod_iter(-1,p**4))
A145297_list = list(islice(A145297_gen(),20)) # Chai Wah Wu, May 04 2024

A145298 Smallest k such that k^2+1 is divisible by A002144(n)^5.

Original entry on oeis.org

1068, 143044, 390112, 7745569, 6423465, 46464143, 23048345, 144762466, 404034898, 2153335831, 331407850, 1108900220, 2581164875, 760839155, 10734466938, 6595297216, 773302059, 61063137802, 31915893786, 112699451831

Offset: 1

Klaus Brockhaus, Oct 14 2008

nonn

			a(4) = 7745569 since A002144(4) = 29, 7745569^2+1 = 59993839133762 = 2*29^5*97*15077 and for no k < 7745569 does 29^5 divide k^2+1.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A002144 (primes of form 4n+1), A002313 (-1 is a square mod p), A059321, A145296, A145297, A145299.

{e=5; forprime(p=2, 200, if(p%4==1, q=p^e; m=q; while(!ispower(m-1,2,&n), m=m+q); print1(n, ",")))}

from itertools import islice
from sympy import nextprime, sqrt_mod_iter
def A145298_gen(): # generator of terms
    p = 1
    while (p:=nextprime(p)):
        if p&3==1:
            yield min(sqrt_mod_iter(-1,p**5))
A145298_list = list(islice(A145298_gen(),20)) # Chai Wah Wu, May 04 2024

A248531 Numbers n such that the smallest prime divisor of n^2+1 is 41.

Original entry on oeis.org

50, 114, 196, 214, 296, 624, 706, 770, 870, 934, 1034, 1180, 1280, 1426, 1444, 1590, 1690, 1754, 1836, 1936, 2000, 2164, 2246, 2264, 2346, 2574, 2674, 2756, 2820, 2984, 3066, 3084, 3230, 3330, 3394, 3494, 3576, 3640, 3740, 3886, 3904, 4214, 4296, 4460, 4624

Offset: 1

Michel Lagneau, Oct 08 2014

Or numbers n such that the smallest prime divisor of n^2+1 is A002313(7).

a(n)== 32 or 50 (mod 82).

			50 is in the sequence because 50^2+1= 41*61.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A089120, A002313, A209874, A248527, A248528, A248529, A248530.

[n: n in [2..5000] | PrimeDivisors(n^2+1)[1] eq 41]; // Bruno Berselli, Oct 08 2014

lst={};Do[If[FactorInteger[n^2+1][[1, 1]]==41, AppendTo[lst, n]], {n, 2, 2000}]; lst
Select[Range[5000],FactorInteger[#^2+1][[1,1]]==41&] (* Harvey P. Dale, Aug 15 2017 *)
p = 41; ps = Select[Range[p - 1], Mod[#, 4] != 3 && PrimeQ[#] &]; Select[Range[5000], Divisible[(nn = #^2 + 1), p] && ! Or @@ Divisible[nn, ps] &] (* Amiram Eldar, Aug 16 2019 *)

A057129 -4 is a square mod n.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 13, 17, 20, 25, 26, 29, 34, 37, 40, 41, 50, 52, 53, 58, 61, 65, 68, 73, 74, 82, 85, 89, 97, 100, 101, 104, 106, 109, 113, 116, 122, 125, 130, 136, 137, 145, 146, 148, 149, 157, 164, 169, 170, 173, 178, 181, 185, 193, 194, 197, 200, 202, 205, 212

Offset: 1

Henry Bottomley, Aug 10 2000

nonn

Numbers that are not multiples of 16 and for which all odd prime factors are congruent to 1 mod 4. - Eric M. Schmidt, Apr 21 2013

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Includes the primes in A002313 and these (primes congruent to {1, 2} mod 4) are the prime factors of the terms in this sequence. Cf. A008784, A057125, A057126, A057127, A057128.

Select[Range[100], IntegerQ[PowerMod[-4, 1/2, #]] &] // Quiet (* After Jean-François Alcover *) (* Robert Price, Apr 19 2025 *)

def A057129(n) :
    if n%16==0: return False
    for (p, m) in factor(n) :
        if p % 4 not in [1, 2] : return False
    return True
# Eric M. Schmidt, Apr 21 2013

A201278 a(n) specifies the quadratic extension sqrt(a(n)) for A201047(n).

Original entry on oeis.org

10, 2, 2, 5, 5, 130, 185, 5, 2, 2, 10, 10, 5, 5, 10, 17, 17, 5, 5, 5, 53, 53, 13, 13, 1490, 5, 2, 2, 5, 1565, 5

Offset: 1

Artur Jasinski, Nov 29 2011

Conjecture (Jasiński): The numbers in this sequence are multiplicative combinations of: primes congruent to 1 or 2 modulo 4 (A002313), Pythagorean primes (A002144), the number 2, and norms of Gaussian primes A055025.

Cf. A200216, A200656, A201047.

Minor edits by N. J. A. Sloane, Feb 23 2014

A079887 Values of y-x where p runs through the primes of form 4k+1 and p=x^2+y^2, 0<=x<=y.

Original entry on oeis.org

1, 1, 3, 3, 5, 1, 5, 1, 5, 3, 5, 9, 7, 1, 7, 3, 5, 11, 1, 5, 13, 13, 5, 11, 15, 3, 5, 11, 15, 1, 3, 7, 13, 9, 11, 7, 13, 19, 17, 1, 5, 13, 17, 9, 17, 9, 11, 5, 7, 23, 15, 19, 1, 3, 21, 9, 19, 11, 25, 21, 7, 25, 17, 1, 13, 5, 15, 23, 11, 17, 5, 25, 23, 9, 3, 5, 19, 15, 27, 25, 13, 1, 19, 29, 27

Offset: 1

Benoit Cloitre, Jan 13 2003

nonn

Also values of x where p runs through the primes of form 4k+1 and 2*p=x^2+y^2, 0<=xColin Barker, Jul 07 2014

Cf. A002330, A002331, A002313, A002144, A079886.

pp = Select[ Range[200] // Prime, Mod[#, 4] == 1 &]; f[p_] := y - x /. ToRules[ Reduce[0 <= x <= y && p == x^2 + y^2, {x, y}, Integers]]; A079887 = f /@ pp (* Jean-François Alcover, Jan 15 2015 *)

a(n) = A002330(n+1)-A002331(n+1). - R. J. Mathar, Jan 09 2017

A112634 Mersenne prime indices that are not Gaussian primes.

Original entry on oeis.org

2, 5, 13, 17, 61, 89, 521, 2281, 3217, 4253, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 132049, 859433, 1398269, 2976221, 3021377, 6972593, 13466917, 30402457, 32582657, 42643801, 43112609, 57885161

Offset: 1

Jorge Coveiro, Dec 27 2005

57885161, 74207281 and 77232917 are in this sequence as well. - Ivan Panchenko, Apr 13 2018
82589933 is in the sequence as well. - David Benjamin, Mar 30 2022
136279841 is in the sequence. - David Benjamin, Nov 11 2024
Other than the term 2, primes p (A000043) such that 2^p - 1 is prime (A000668) and congruent to 31 mod 120. - Jianing Song, Nov 18 2024

Chris K. Caldwell, The largest known primes.

Cf. A000043, A112633, A002313.

Select[MersennePrimeExponent[Range[48]], Mod[#, 4] != 3 &] (* Amiram Eldar, Oct 18 2024 *)

is(n)=n%4 < 3 && isprime(n) && isprime(2^n-1) \\ Charles R Greathouse IV, Nov 28 2016

A000043 INTERSECT A002313. - R. J. Mathar, Oct 06 2008

A000043 SET-MINUS A112633.

Edited by R. J. Mathar, Oct 06 2008
a(26)-a(28) from Ivan Panchenko, Apr 13 2018
a(29) from Amiram Eldar, Oct 18 2024

A137351 Composite numbers n such that x^2 - n*y^2 represents -1.

Original entry on oeis.org

10, 26, 50, 58, 65, 74, 82, 85, 106, 122, 125, 130, 145, 170, 185, 202, 218, 226, 250, 265, 274, 290, 298, 314, 325, 338, 346, 362, 365, 370, 394, 425, 442, 445, 458, 481, 485, 493, 530, 533, 538, 554, 565, 586, 610, 626, 629, 634, 685, 697, 698, 730, 746

Offset: 1

N. J. A. Sloane, Apr 08 2008

nonn

Number of terms less than or equal to 10^k for k=0 .. : 0, 1, 8, 71, 712, 6702, 63485, 602870, ... . - Robert G. Wilson v, Jul 20 2008

			3^2 - 10*1^2 = -1, so 10 is a member.
4005^2 - 106*389^2 = -1, so 106 is a member.

Robert G. Wilson v, Table of n, a(n) for n = 1..63485
J. P. Robertson and K. R. Matthews, A continued fraction approach to a result of Feit, Amer. Math. Monthly, 115 (No. 4, 2008), 346-349.
Eric Weisstein's World of Mathematics, Pell Equation.

For the primes with this property see A002313. A134406 is a subset.

lst = {}; Do[ If[ !PrimeQ@ n && FindInstance[x^2 - n*y^2 == -1, {x, y}, Integers] != {}, AppendTo[lst, n]], {n, 2, 1000}]

More terms from Robert G. Wilson v, Jul 20 2008

A248528 Numbers n such that the smallest prime divisor of n^2+1 is 17.

Original entry on oeis.org

4, 30, 64, 106, 140, 166, 234, 276, 310, 336, 344, 370, 404, 446, 480, 506, 514, 540, 574, 650, 676, 744, 786, 820, 846, 854, 880, 914, 956, 990, 1016, 1024, 1050, 1160, 1186, 1194, 1220, 1254, 1296, 1330, 1356, 1364, 1390, 1424, 1466, 1534, 1560, 1636, 1670

Offset: 1

Michel Lagneau, Oct 08 2014

Or numbers n such that the smallest prime divisor of n^2+1 is A002313(4).

a(n)== 4 or 30 (mod 34).

			30 is in the sequence because 30^2+1= 17*53.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A089120, A002313, A248527.

[n: n in [2..2000] | PrimeDivisors(n^2+1)[1] eq 17]; // Bruno Berselli, Oct 08 2014

lst={};Do[If[FactorInteger[n^2+1][[1, 1]]==17, AppendTo[lst, n]], {n, 2, 2000}]; lst
p = 17; ps = Select[Range[p - 1], Mod[#, 4] != 3 && PrimeQ[#] &]; Select[Range[1670], Divisible[(nn = #^2 + 1), p] && ! Or @@ Divisible[nn, ps] &] (* Amiram Eldar, Aug 16 2019 *)

A248549 Numbers n such that the smallest prime divisor of n^2+1 is 61.

Original entry on oeis.org

194, 316, 416, 804, 904, 926, 1026, 1170, 1270, 1414, 1536, 1780, 2024, 2490, 2734, 2856, 3000, 3100, 3244, 3344, 3366, 3610, 3954, 3976, 4320, 4564, 4830, 4930, 5074, 5196, 5540, 5684, 6394, 6416, 6516, 6760, 6904, 7004, 7126, 7270, 7370, 7514, 7614, 7736

Offset: 1

Michel Lagneau, Oct 08 2014

Or numbers n such that the smallest prime divisor of n^2+1 is A002313(9).

a(n)== 50 or 72 (mod 122).

			194 is in the sequence because 194^2+1= 61*617.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A089120, A002313, A209874, A248527-A248531, A248549-A248553.

lst={};Do[If[FactorInteger[n^2+1][[1, 1]]==61, AppendTo[lst, n]], {n, 2, 10000}]; lst
p = 61; ps = Select[Range[p - 1], Mod[#, 4] != 3 && PrimeQ[#] &]; Select[Range[8000], Divisible[(nn = #^2 + 1), p] && ! Or @@ Divisible[nn, ps] &] (* Amiram Eldar, Aug 16 2019 *)

cp's OEIS Frontend

A145297 Smallest k such that k^2+1 is divisible by A002144(n)^4.

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A145298 Smallest k such that k^2+1 is divisible by A002144(n)^5.

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A248531 Numbers n such that the smallest prime divisor of n^2+1 is 41.

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A057129 -4 is a square mod n.

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A201278 a(n) specifies the quadratic extension sqrt(a(n)) for A201047(n).

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A079887 Values of y-x where p runs through the primes of form 4k+1 and p=x^2+y^2, 0<=x<=y.

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Mathematica

Formula

A112634 Mersenne prime indices that are not Gaussian primes.

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Mathematica

PARI

Formula

Extensions

A137351 Composite numbers n such that x^2 - n*y^2 represents -1.

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Mathematica