A084926 Inverse hyperbolic cotangent reducible numbers. The hyperbolic analog of the non-Stormer numbers (A002312).
5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 56, 57, 59, 61, 63, 64, 65, 67, 69, 71, 73, 75, 76, 77, 79, 81, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 116, 117, 118
Offset: 1
Keywords
A005528 Størmer numbers or arc-cotangent irreducible numbers: numbers k such that the largest prime factor of k^2 + 1 is >= 2*k.
1, 2, 4, 5, 6, 9, 10, 11, 12, 14, 15, 16, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 33, 34, 35, 36, 37, 39, 40, 42, 44, 45, 48, 49, 51, 52, 53, 54, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 71, 74, 77, 78, 79, 80, 81, 82, 84, 85, 86, 87, 88, 89, 90, 92, 94, 95, 96
Offset: 1
Comments
Also numbers k such that k^2 + 1 has a primitive divisor, hence (by Everest & Harman, Theorem 1.4) 1.1n < a(n) < 1.88n for large enough n. They conjecture that a(n) ~ cn where c = 1/log 2 = 1.4426.... - Charles R Greathouse IV, Nov 15 2014
Named after the Norwegian mathematician and astrophysicist Carl Størmer (1874-1957). - Amiram Eldar, Jun 08 2021
References
- John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, p. 246.
- 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 and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- John Todd, Table of Arctangents, National Bureau of Standards, Washington, DC, 1951, p. 2.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
- Graham Everest and Glyn Harman, On primitive divisors of n^2 + b, arXiv:math/0701234 [math.NT], 2007.
- Carl Størmer, Sur l'application de la théorie des nombres entiers complexes à la solution en nombres rationnels x_1 x_2... x_n c_1 c_2... c_n, k de l'équation: c_1 arc tg x_1 + c_2 arc tg x_2 + ... + c_n arc tg x_n = k * Pi/4, Archiv for mathematik og naturvidenskab, Vol. 19, No. 3 (1896), pp. 1-96.
- John Todd, A problem on arc tangent relations, Amer. Math. Monthly, 56 (1949), 517-528.
- Eric Weisstein's World of Mathematics, Størmer Number.
- Wikipedia, Størmer number.
Programs
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Haskell
a005528 n = a005528_list !! (n-1) a005528_list = filter (\x -> 2 * x <= a006530 (x ^ 2 + 1)) [1..] -- Reinhard Zumkeller, Jun 12 2015
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Mathematica
Select[Range[96], FactorInteger[#^2 + 1][[-1, 1]] >= 2 # &] (* Jean-François Alcover, Apr 11 2011 *)
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PARI
is(n)=my(f=factor(n^2+1)[,1]);f[#f]>=2*n \\ Charles R Greathouse IV, Nov 14 2014
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Python
from sympy import factorint def ok(n): return max(factorint(n*n + 1)) >= 2*n print(list(filter(ok, range(1, 97)))) # Michael S. Branicky, Aug 30 2021
A005529 Primitive prime factors of the sequence k^2 + 1 (A002522) in the order that they are found.
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
Keywords
Comments
Primes associated with Stormer numbers.
References
- 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.
Links
- 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
Programs
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Magma
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
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Mathematica
prms={}; Do[f=First/@FactorInteger[k^2+1]; p=Complement[f, prms]; prms=Join[prms, p], {k, 100}]; prms -
PARI
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
Extensions
Edited by T. D. Noe, Oct 02 2003
A071931 Non-Størmer numbers whose largest prime factor is a Størmer number.
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
Keywords
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Programs
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Haskell
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
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Python
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
A260258 T(n,k) is the array read by rows, n>0 and k=1..q (with q = number of prime distinct divisors of n^2+1) giving the number of occurrences of the k-th prime divisor of n^2+1 counted from the prime divisors of m^2+1 for m=1..n.
1, 1, 2, 2, 1, 3, 1, 1, 4, 3, 4, 2, 5, 1, 1, 6, 1, 5, 1, 7, 6, 2, 1, 8, 1, 1, 9, 7, 2, 8, 3, 10, 1, 1, 11, 4, 3, 9, 1, 12, 10, 1, 1, 13, 1, 1, 14, 11, 1, 12, 1, 15, 1, 4, 2, 16, 5, 2, 13, 2, 17, 14, 1, 6, 1, 18, 1, 1, 19, 15, 1, 16, 5, 20, 1, 1, 21, 3, 17, 1
Offset: 1
Comments
Examples
T(13,k) = [7,6,2] for k = 1,2,3 because 13^2+1 = 2*5*17 => The number of occurrences of the prime divisor 2 is 7: 1^2+1=2, 3^2+1=2*5, 5^2+1=2*13, 7^2+1=2*5^2, 9^2+1=2*41, 11^2+1=2*61 and 13^2+1=2*5*17; The number of occurrences of the prime divisor 5 is 6: 2^2+1=5, 3^2+1=2*5, 7^2+1=2*5^2, 8^2+1=5*13, 12^2+1=5*29; The number of occurrences of the prime divisor 17 is 2: 4^2+1=17 and 13^2+1=2*5*17. The array begins: [1] [1] [2,2] [1] [3,1] [1] [4,3] [4,2] [5,2] [1] ...
Links
- Michel Lagneau, Table of n, a(n) for n = 1..5000
Programs
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Maple
with(numtheory):lst:={2}:nn:=1000:T:=array(1..270,[0$270]): for j from 1 to nn do: p:=4*j+1: if isprime(p) then lst:=lst union {p}: fi: od: nn0:=nops(lst): for n from 1 to 60 do: q:=factorset(n^2+1):n0:=nops(q): for k from 1 to n0 do: for m from 1 to 270 do: if q[k]=lst[m] then T[m]:=T[m]+1:printf(`%d, `, T[m]): fi: od: od: od:
A120294 Numerator of determinant of n X n matrix with elements M[j,j] = (i+j)/(i+j-1).
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
Comments
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.
Programs
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Mathematica
Numerator[Table[Det[Table[(i+j)/(i+j-1),{i,1,n},{j,1,n}]],{n,1,40}]]
Formula
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
7, 47, 57, 99, 117
Offset: 1
Comments
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.
Examples
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.
Links
- Eric Weisstein's MathWorld, Inverse Tangent.
Comments
Crossrefs
Programs
PARI