A005528 -id:A005528 - cp's OEIS frontend

A002314 Minimal integer square root of -1 modulo p, where p is the n-th prime of the form 4k+1.

2, 5, 4, 12, 6, 9, 23, 11, 27, 34, 22, 10, 33, 15, 37, 44, 28, 80, 19, 81, 14, 107, 89, 64, 16, 82, 60, 53, 138, 25, 114, 148, 136, 42, 104, 115, 63, 20, 143, 29, 179, 67, 109, 48, 208, 235, 52, 118, 86, 24, 77, 125, 35, 194, 154, 149, 106, 58, 26, 135, 96, 353, 87, 39

Offset: 1

N. J. A. Sloane

nonn

In other words, if p is the n-th prime == 1 (mod 4), a(n) is the smallest positive integer k such that k^2 + 1 == 0 (mod p).
The 4th roots of unity mod p, where p = n-th prime == 1 (mod 4), are +1, -1, a(n) and p-a(n).
Related to Stormer numbers.
Comment from Igor Shparlinski, Mar 12 2007 (writing to the Number Theory List): Results about the distribution of roots (for arbitrary quadratic polynomials) are given by W. Duke, J. B. Friedlander and H. Iwaniec and A. Toth.
Comment from Emmanuel Kowalski, Mar 12 2007 (writing to the Number Theory List): It is known (Duke, Friedlander, Iwaniec, Annals of Math. 141 (1995)) that the fractional part of a(n)/p(n) is equidistributed in [0,1/2] for p(n)From Artur Jasinski, Dec 10 2008: (Start)
If we take the four numbers 1, A002314(n), A152676(n), and A152680(n), then their multiplication table modulo A002144(n) is isomorphic to the Latin square
    1 2 3 4
    2 4 1 3
    3 1 4 2
    4 3 2 1
and isomorphic to the multiplication table of {1, i, -i, -1} where i=sqrt(-1), A152680(n) is isomorphic to -1, A002314(n) with i or -i and A152676(n) vice versa -i or i.
1, A002314(n), A152676(n), A152680(n) are subfield of Galois Field [A002144(n)]. (End)
It is found empirically that the solutions of the Diophantine equation X^4 + Y^2 == 0 (mod P) (where P is a prime of the form P=4k+1) are integer points on parabolas Y = (+-(X^2 - P*X) + P*i)/C(P) where C(P) is the term corresponding to a prime P in this sequence. - Seppo Mustonen, Sep 22 2020

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. See Vol. 1, pp. 1-21.
W. Duke, J. B. Friedlander and H. Iwaniec, Equidistribution of roots of a quadratic congruence to prime moduli, Annals of Math, 141 (1995), 423-441.
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. 521.
Seppo Mustonen, Roots of Diophantine equations mod(X^4+Y^2,P)=0, 2020
Seppo Mustonen, Roots of Diophantine equations mod(X^4+Y^2,P)=0, Youtube video, 2020
J. Todd, A problem on arc tangent relations, Amer. Math. Monthly, 56 (1949), 517-528.
A. Toth, Roots of quadratic congruences, Intern. Math. Research Notices, 2000 (2000), 719-739.

Cf. A002313, A005528, A047818, A002144, A152676, A152680. Subsequence of A057756.

f:=proc(n) local i,j,k; for i from 1 to (n-1)/2 do if i^2 +1 mod n = 0 then RETURN(i); fi od: -1; end;
t1:=[]; M:=40; for n from 1 to M do q:=ithprime(n); if q mod 4 = 1 then t1:=[op(t1),f(q)]; fi; od: t1;

aa = {}; Do[If[Mod[Prime[n], 4] == 1, k = 1; While[ ! Mod[k^2 + 1, Prime[n]] == 0, k++ ]; AppendTo[aa, k]], {n, 1, 100}]; aa (* Artur Jasinski, Dec 10 2008 *)

first_N_terms(N) = my(v=vector(N), i=0); forprime(p=5, oo, if(p%4==1, i++; v[i] = lift(sqrt(Mod(-1,p)))); if(i==N, break())); v \\ Jianing Song, Apr 17 2021

Better description from Tony Davie (ad(AT)dcs.st-and.ac.uk), Feb 07 2001

More terms from Jud McCranie, Mar 18 2001

A005529 Primitive prime factors of the sequence k^2 + 1 (A002522) in the order that they are found.

Original entry on oeis.org

2, 5, 17, 13, 37, 41, 101, 61, 29, 197, 113, 257, 181, 401, 97, 53, 577, 313, 677, 73, 157, 421, 109, 89, 613, 1297, 137, 761, 1601, 353, 149, 1013, 461, 1201, 1301, 541, 281, 2917, 3137, 673, 1741, 277, 1861, 769, 397, 241, 2113, 4357, 449, 2381, 2521, 5477

Offset: 1

N. J. A. Sloane

nonn

Primes associated with Stormer numbers.

See A002313 for the sorted list of primes. It can be shown that k^2 + 1 has at most one primitive prime factor; the other prime factors divide m^2 + 1 for some m < k. When k^2 + 1 has a primitive prime factor, k is a Stormer number (A005528), otherwise a non-Stormer number (A002312).

John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, p. 246.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. Todd, Table of Arctangents. National Bureau of Standards, Washington, DC, 1951, p. vi.

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Stormer Number.
Eric Weisstein's World of Mathematics, Primitive Prime Factor

Cf. A002312, A002313 (primes of the form 4k+1), A002522, A005528.

V:=[]; for n in [1..75] do p:=Max([ x[1]: x in Factorization(n^2+1) ]); if not p in V then Append(~V, p); end if; end for; V; // Klaus Brockhaus, Oct 29 2008

prms={}; Do[f=First/@FactorInteger[k^2+1]; p=Complement[f, prms]; prms=Join[prms, p], {k, 100}]; prms

do(n)=my(v=List(),g=1,m,t,f); for(k=1,n, m=k^2+1; t=gcd(m,g); while(t>1, m/=t; t=gcd(m,t)); f=factor(m)[,1]; if(#f, listput(v,f[1]); g*=f[1])); Vec(v) \\ Charles R Greathouse IV, Jun 11 2017

Edited by T. D. Noe, Oct 02 2003

A002312 Arc-cotangent reducible numbers or non-Størmer numbers: largest prime factor of k^2 + 1 is less than 2*k.

Original entry on oeis.org

3, 7, 8, 13, 17, 18, 21, 30, 31, 32, 38, 41, 43, 46, 47, 50, 55, 57, 68, 70, 72, 73, 75, 76, 83, 91, 93, 98, 99, 100, 105, 111, 112, 117, 119, 122, 123, 128, 129, 132, 133, 142, 144, 155, 157, 162, 172, 173, 174, 177, 182, 183, 185, 187, 189, 191, 192, 193, 200

Offset: 1

N. J. A. Sloane

Also numbers such that k^2 + 1 has no primitive divisor, hence (by Everest & Harman, Theorem 1.4) 2.138n < a(n) < 10.6n for large enough n. They conjecture that a(n) ~ cn where c = 1/(1 - log 2) = 3.258.... - Charles R Greathouse IV, Nov 15 2014

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 247.
Graham Everest and Glyn Harman, On primitive divisors of n^2 + b, in Number Theory and Polynomials (James McKee and Chris Smyth, ed.), London Mathematical Society 2008.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. Todd, Table of Arctangents. National Bureau of Standards, Washington, DC, 1951, p. 94.

T. D. Noe, Table of n, a(n) for n = 1..1000
Graham Everest and Glyn Harman, On primitive divisors of n^2 + b, arXiv:math/0701234 [math.NT], 2007.
E. Kowalski, On the "reducibility" of arctangents of integers, Amer. Math. Monthly, Vol. 111, No. 4 (Apr. 2004), 351-354.
Olga Taussky, Sums of Squares, The American Mathematical Monthly, Vol. 77, No. 8 (Oct., 1970), pp. 805-830 (26 pages). See p. 823.
J. Todd, A problem on arc tangent relations, Amer. Math. Monthly, 56 (1949), 517-528.

Cf. A005528.

Cf. A006530, A071931 (subsequence).

a002312 n = a002312_list !! (n-1)
a002312_list = filter (\x -> 2 * x > a006530 (x ^ 2 + 1)) [1..]
-- Reinhard Zumkeller, Jun 12 2015

lst={}; Do[n=m^2+1; p=FactorInteger[n][[ -1, 1]]; If[p<2m, AppendTo[lst, m]], {m, 200}]; lst (* T. D. Noe, Apr 09 2004 *)
Select[Range[200],FactorInteger[#^2+1][[-1,1]]<2#&] (* Harvey P. Dale, Dec 07 2015 *)

is(n)=my(f=factor(n^2+1)[,1]);f[#f]<2*n \\ Charles R Greathouse IV, Nov 14 2014

from sympy import factorint
def ok(n): return max(factorint(n*n + 1)) < 2*n
print(list(filter(ok, range(1, 201)))) # Michael S. Branicky, Aug 30 2021

Description and initial term modified Jan 15 1996

More terms from Jason Earls, Jun 14 2002

A071931 Non-Størmer numbers whose largest prime factor is a Størmer number.

Original entry on oeis.org

8, 30, 32, 38, 46, 50, 55, 57, 75, 76, 99, 100, 111, 122, 128, 132, 133, 142, 174, 177, 183, 185, 200, 203, 212, 213, 228, 237, 242, 253, 254, 265, 266, 268, 274, 278, 302, 305, 319, 322, 327, 334, 342, 348, 360, 377, 380, 381, 394, 395, 411, 413, 418, 437

Offset: 1

Jason Earls, Jun 14 2002

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A005528, A002312.

Cf. A006530.

a071931 n = a071931_list !! (n-1)
a071931_list = filter f a002312_list where
   f x = 2 * gpf <= a006530 (gpf ^ 2 + 1) where gpf = a006530 x
-- Reinhard Zumkeller, Jun 12 2015

from sympy import factorint
def stormer(n): return max(factorint(n*n + 1)) >= 2*n
def ok(n): return not stormer(n) and stormer(max(factorint(n)))
print(list(filter(ok, range(1, 438)))) # Michael S. Branicky, Aug 30 2021

A047818 a(n) is the least number m such that A002313(n)*m - 1 is a square.

Original entry on oeis.org

1, 1, 2, 1, 5, 1, 2, 10, 2, 10, 13, 5, 1, 10, 2, 10, 13, 5, 37, 2, 34, 1, 50, 34, 17, 1, 25, 13, 10, 65, 2, 41, 65, 53, 5, 29, 34, 10, 1, 50, 2, 74, 10, 26, 5, 85, 106, 5, 25, 13, 1, 10, 26, 2, 61, 37, 34, 17, 5, 1, 26, 13, 170, 10, 2, 5, 130, 58, 125, 106, 73, 130, 50, 26, 170

Offset: 1

N. J. A. Sloane

nonn

A002313 has the 4k+1 and 4k+2 primes.

Related to Stormer numbers.

			a(3) = 2 because A002313(3)=13 and 13*2-1 = 5^2.

J. Todd, A problem on arc tangent relations, Amer. Math. Monthly, 56 (1949), 517-528.

Cf. A002313, A002314, A005528.

a(n) = ((A002314(n-1))^2 + 1) / A002313(n).

Edited by Don Reble, Apr 13 2006

A261345 Number of distinct prime divisors among the numbers k^2 + 1 for k in 1 <= k <= n.

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 5, 5, 6, 7, 8, 9, 9, 10, 11, 12, 12, 12, 13, 14, 14, 15, 16, 17, 18, 19, 20, 21, 22, 22, 22, 22, 23, 24, 25, 26, 27, 27, 28, 29, 29, 30, 30, 31, 32, 32, 32, 33, 34, 34, 35, 36, 37, 38, 38, 39, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 49, 50

Offset: 1

Michel Lagneau, Aug 15 2015

nonn

Conjecture: n/a(n) <= 1.6.

Størmer-number-counting function: a(n) is the number of terms in A005528 less than or equal to n. - Luc Rousseau, Jun 13 2018

			For a(5), there are 4 distinct prime divisors that occur in the values 1^2+1 = 2, 2^2+1 = 5, 3^2+1 = 2*5, 4^2+1 = 17, 5^2+1 = 26 = 2*13. Taken together, the distinct prime factors are {2,5,13,17}.

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A002144, A002496, A002522, A089122, A005528.

with(numtheory):nn:=100:lst:={}:
for n from 1 to nn do:
  p:=n^2+1:x:=factorset(p):n0:=nops(x):
  A:={op(x),x[n0]}:
  lst:=lst union A :n1:=nops(lst):printf(`%d, `,n1):
od:

Array[Length@ Tally@ First@ Transpose@ Flatten[FactorInteger[#^2 + 1] & /@ Range@ #, 1] &, {69}] (* Michael De Vlieger, Aug 18 2015 *)
Module[{nn=70,fi},fi=Table[FactorInteger[n^2+1][[All,1]],{n,nn}];Table[ Length[ Union[Flatten[Take[fi,m]]]],{m,nn}]] (* Harvey P. Dale, Sep 11 2021 *)

lista(nn) = {v = []; for (n=1, nn, v = Set(concat(v, factor(n^2+1)[,1]~)); print1(#v, ", "););} \\ Michel Marcus, Aug 16 2015

A120294 Numerator of determinant of n X n matrix with elements M[j,j] = (i+j)/(i+j-1).

Original entry on oeis.org

2, 5, 1, 17, 13, 37, 1, 1, 41, 101, 61, 29, 1, 197, 113, 257, 1, 1, 181, 401, 1, 97, 53, 577, 313, 677, 73, 157, 421, 1, 1, 1, 109, 89, 613, 1297, 137, 1, 761, 1601

Offset: 1

Alexander Adamchuk, Jul 10 2006

Some a(n) are equal to 1 (for n=3,7,8,13,17,18,21,30,31,32,38..=A002312 Arc-cotangent reducible numbers or non-Stormer numbers). All other a(n) (for n=1,2,4,5,6,9,10,11,14,15,16,19,20,22,23..=A005528 Stormer numbers or arc-cotangent irreducible numbers, largest prime factor of n^2 + 1 is >= 2n.) belong to A005529 - Primitive prime factors of the sequence k^2 + 1 (A002522) in the order that they are found. Matrix M[i,j] = (i+j)/(i+j-1) = 1 + 1/(i+j-1) is a sum of n X n unit matrix and n X n Hilbert Matrix. Denominator of determinant of matrix M[i,j] equals determinant of inverse Hilbert matrix A005249.

Cf. A005249, A002312, A005528, A005529, A002522.

Numerator[Table[Det[Table[(i+j)/(i+j-1),{i,1,n},{j,1,n}]],{n,1,40}]]

a(n) = numerator[Det[Table[(i+j)/(i+j-1),{i,1,n},{j,1,n}]]].

A263920 A positive integer n is in this sequence iff arctan(n)^2 can be represented as Sum_{0

Original entry on oeis.org

7, 47, 57, 99, 117

Offset: 1

Vladimir Reshetnikov, Oct 29 2015

The terms given are certainly in the sequence. Although I lack a rigorous proof that no intermediate terms were omitted, an extensive computer search gave no other candidates in between.

It is an open question if the sequence is infinite.

			7 is in the sequence, because arctan(7)^2 = -5*arctan(1)^2 + (10/3)*arctan(2)^2 + (2/3)*arctan(3)^2.
47 is in the sequence, because arctan(47)^2 = (2939/210)*arctan(2)^2 - (125/21)*arctan(3)^2 - (6/5)*arctan(4)^2 - (12/7)*arctan(5)^2 - (29/7)*arctan(7)^2 + (15/7)*arctan(8)^2 + (2/5)*arctan(13)^2 + (11/7)*arctan(18)^2 - arctan(21)^2 + (7/10)*arctan(38)^2.

Eric Weisstein's MathWorld, Inverse Tangent.

Cf. A005528, A002312.

Showing 1-9 of 9 results.

n is in the sequence if y = (xn+1)/(x+n) is noninteger for all integer x where 1 < x < n. Equivalently, n is in the sequence when n cannot be formed by (xy-1)/(x-y) for all integers x and y where x < n and 1 < y < x, so n cannot satisfy ((n+1)/(n-1))*((x+1)/(x-1)) = ((y+1)/(y-1)). Thus all the nearest neighbors of the primes (A045718) appear in this sequence.

cp's OEIS Frontend

A084925 Inverse hyperbolic cotangent irreducible numbers: positive integers such that the arccoth of these numbers form a basis for the space of arccoth of rationals >=1. The hyperbolic analog of the Stormer numbers (A005528).

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A002314 Minimal integer square root of -1 modulo p, where p is the n-th prime of the form 4k+1.

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A005529 Primitive prime factors of the sequence k^2 + 1 (A002522) in the order that they are found.

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A002312 Arc-cotangent reducible numbers or non-Størmer numbers: largest prime factor of k^2 + 1 is less than 2*k.

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A071931 Non-Størmer numbers whose largest prime factor is a Størmer number.

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A047818 a(n) is the least number m such that A002313(n)*m - 1 is a square.

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A261345 Number of distinct prime divisors among the numbers k^2 + 1 for k in 1 <= k <= n.

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