A059592 -id:A059592 - cp's OEIS frontend

A002522 a(n) = n^2 + 1.

1, 2, 5, 10, 17, 26, 37, 50, 65, 82, 101, 122, 145, 170, 197, 226, 257, 290, 325, 362, 401, 442, 485, 530, 577, 626, 677, 730, 785, 842, 901, 962, 1025, 1090, 1157, 1226, 1297, 1370, 1445, 1522, 1601, 1682, 1765, 1850, 1937, 2026, 2117, 2210, 2305, 2402, 2501

Offset: 0

N. J. A. Sloane

An n X n nonnegative matrix A is primitive (see A070322) iff every element of A^k is > 0 for some power k. If A is primitive then the power which should have all positive entries is <= n^2 - 2n + 2 (Wielandt).
a(n) = Phi_4(n), where Phi_k is the k-th cyclotomic polynomial.
As the positive solution to x=2n+1/x is x=n+sqrt(a(n)), the continued fraction expansion of sqrt(a(n)) is {n; 2n, 2n, 2n, 2n, ...}. - Benoit Cloitre, Dec 07 2001
a(n) is one less than the arithmetic mean of its neighbors: a(n) = (a(n-1) + a(n+1))/2 - 1. E.g., 2 = (1+5)/2 - 1, 5 = (2+10)/2 - 1. - Amarnath Murthy, Jul 29 2003
Equivalently, the continued fraction expansion of sqrt(a(n)) is (n;2n,2n,2n,...). - Franz Vrabec, Jan 23 2006
Number of {12,1*2*,21}-avoiding signed permutations in the hyperoctahedral group.
The number of squares of side 1 which can be drawn without lifting the pencil, starting at one corner of an n X n grid and never visiting an edge twice is n^2-2n+2. - Sébastien Dumortier, Jun 16 2005
Also, numbers m such that m^3 - m^2 is a square, (n*(1 + n^2))^2. - Zak Seidov
1 + 2/2 + 2/5 + 2/10 + ... = Pi*coth Pi [Jolley], see A113319. - Gary W. Adamson, Dec 21 2006
For n >= 1, a(n-1) is the minimal number of choices from an n-set such that at least one particular element has been chosen at least n times or each of the n elements has been chosen at least once. Some games define "matches" this way; e.g., in the classic Parker Brothers, now Hasbro, board game Risk, a(2)=5 is the number of cards of three available types (suits) required to guarantee at least one match of three different types or of three of the same type (ignoring any jokers or wildcards). - Rick L. Shepherd, Nov 18 2007
Positive X values of solutions to the equation X^3 + (X - 1)^2 + X - 2 = Y^2. To prove that X = n^2 + 1: Y^2 = X^3 + (X - 1)^2 + X - 2 = X^3 + X^2 - X - 1 = (X - 1)(X^2 + 2X + 1) = (X - 1)*(X + 1)^2 it means: (X - 1) must be a perfect square, so X = n^2 + 1 and Y = n(n^2 + 2). - Mohamed Bouhamida, Nov 29 2007
{a(k): 0 <= k < 4} = divisors of 10. - Reinhard Zumkeller, Jun 17 2009
Appears in A054413 and A086902 in relation to sequences related to the numerators and denominators of continued fractions convergents to sqrt((2*n)^2/4 + 1), n=1, 2, 3, ... . - Johannes W. Meijer, Jun 12 2010
For n > 0, continued fraction [n,n] = n/a(n); e.g., [5,5] = 5/26. - Gary W. Adamson, Jul 15 2010
The only real solution of the form f(x) = A*x^p with negative p which satisfies f^(m)(x) = f^[-1](x), x >= 0, m >= 1, with f^(m) the m-th derivative and f^[-1] the compositional inverse of f, is obtained for m=2*n, p=p(n)= -(sqrt(a(n))-n) and A=A(n)=(fallfac(p(n),2*n))^(-p(n)/(p(n)+1)), with fallfac(x,k):=Product_{j=0..k-1} (x-j) (falling factorials). See the T. Koshy reference, pp. 263-4 (there are also two solutions for positive p, see the corresponding comment in A087475). - Wolfdieter Lang, Oct 21 2010
n + sqrt(a(n)) = [2*n;2*n,2*n,...] with the regular continued fraction with period 1. This is the even case. For the general case see A087475 with the Schroeder reference and comments. For the odd case see A078370.
a(n-1) counts configurations of non-attacking bishops on a 2 X n strip [Chaiken et al., Ann. Combin. 14 (2010) 419]. - R. J. Mathar, Jun 16 2011
Also numbers k such that 4*k-4 is a square. Hence this sequence is the union of A053755 and A069894. - Arkadiusz Wesolowski, Aug 02 2011
a(n) is also the Moore lower bound on the order, A191595(n), of an (n,5)-cage. - Jason Kimberley, Oct 17 2011
Left edge of the triangle in A195437: a(n+1) = A195437(n,0). - Reinhard Zumkeller, Nov 23 2011
If h (5,17,37,65,101,...) is prime is relatively prime to 6, then h^2-1 is divisible by 24. - Vincenzo Librandi, Apr 14 2014
The identity (4*n^2+2)^2 - (n^2+1)*(4*n)^2 = 4 can be written as A005899(n)^2 - a(n)*A008586(n)^2 = 4. - Vincenzo Librandi, Jun 15 2014
a(n) is also the number of permutations simultaneously avoiding 213 and 321 in the classical sense which can be realized as labels on an increasing strict binary tree with 2n-1 nodes. See A245904 for more information on increasing strict binary trees. - Manda Riehl, Aug 07 2014
a(n-1) is the maximum number of stages in the Gale-Shapley algorithm for finding a stable matching between two sets of n elements given an ordering of preferences for each element (see Gura et al.). - Melvin Peralta, Feb 07 2016
Because of Fermat's little theorem, a(n) is never divisible by 3. - Altug Alkan, Apr 08 2016
For n > 0, if a(n) points are placed inside an n X n square, it will always be the case that at least two of the points will be a distance of sqrt(2) units apart or less. - Melvin Peralta, Jan 21 2017
Also the limit as q->1^- of the unimodal polynomial (1-q^(n*k+1))/(1-q) after making the simplification k=n. The unimodal polynomial is from O'Hara's proof of unimodality of q-binomials after making the restriction to partitions of size <= 1. See G_1(n,k) from arXiv:1711.11252. As the size restriction s increases, G_s->G_infinity=G: the q-binomials. Then substituting k=n and q=1 yields the central binomial coefficients: A000984. - Bryan T. Ek, Apr 11 2018
a(n) is the smallest number congruent to both 1 (mod n) and 2 (mod n+1). - David James Sycamore, Apr 04 2019
a(n) is the number of permutations of 1,2,...,n+1 with exactly one reduced decomposition. - Richard Stanley, Dec 22 2022
From Klaus Purath, Apr 03 2025: (Start)
The odd prime factors of these terms are always of the form 4*k + 1.
All a(n) = D satisfy the Pell equation (k*x)^2 - D*y^2 = -1. The values for k and the solutions x, y can be calculated using the following algorithm: k = n, x(0) = 1, x(1) = 4*D - 1, y(0) = 1, y(1) = 4*D - 3. The two recurrences are of the form (4*D - 2, -1). The solutions x, y of the Pell equations for n = {1 ... 14} are in OEIS.
It follows from the above that this sequence is a subsequence of A031396. (End)

			G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 17*x^4 + 26*x^5 + 37*x^6 + 50*x^7 + 65*x^8 + ...

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 120).
E. Gura and M. Maschler, Insights into Game Theory: An Alternative Mathematical Experience, Cambridge, 2008; p. 26.
Thomas Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley and Sons, New York, 2001.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000. Format corrected by _Peter Kagey_, Jan 25 2016
Wawrzyniec Bieniawski, Piotr Masierak, Andrzej Tomski, and Szymon Łukaszyk, On the Salient Regularities of Strings of Assembly Theory, Preprints (2024). See p. 19.
Wawrzyniec Bieniawski, Piotr Masierak, Andrzej Tomski, and Szymon Łukaszyk, Assembly Theory - Formalizing Assembly Spaces and Discovering Patterns and Bounds, Preprints.org (2025). See p. 28.
R. P. Boas and N. J. A. Sloane, Correspondence, 1974
Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 44, 56.
Giulio Cerbai and Luca Ferrari, Permutation patterns in genome rearrangement problems: the reversal model, arXiv:1903.08774 [math.CO], 2019. See p. 19.
S. Chaiken et al., Nonattacking Queens in a Rectangular Strip, arXiv:1105.5087 [math.CO], 2011.
Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018.
R. M. Green and Tianyuan Xu, 2-roots for simply laced Weyl groups, arXiv:2204.09765 [math.RT], 2022.
Guo-Niu Han, Enumeration of Standard Puzzles
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv:1410.2657 [math.CO], 2014.
C. Homberger and V. Vatter, On the effective and automatic enumeration of polynomial permutation classes, arXiv:1308.4946 [math.CO], 2013.
L. B. W. Jolley, Summation of Series, Dover, 1961, p. 176.
S. J. Leon, Linear Algebra with Applications: the Perron-Frobenius Theorem [Cached copy at the Wayback Machine]
T. Mansour and J. West, Avoiding 2-letter signed patterns, arXiv:math/0207204 [math.CO], 2002.
Michelle Rudolph-Lilith, On the Product Representation of Number Sequences, with Application to the Fibonacci Family, arXiv:1508.07894 [math.NT], 2015.
Eric Weisstein's World of Mathematics, Number Picking
Eric Weisstein's World of Mathematics, Near-Square Prime
Helmut Wielandt, Unzerlegbare, nicht negative Matrizen, Math. Z. 52 (1950), 642-648. volume 52
Reinhard Zumkeller, Enumerations of Divisors
Index to values of cyclotomic polynomials of integer argument
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Left edge of A055096.
Cf. A059100, A117950, A087475, A117951, A114949, A117619 (sequences of form n^2 + K).
a(n+1) = A101220(n, n+1, 3).
Cf. A059592, A124808, A132411, A132414, A028872, A005408, A000124, A016813, A086514, A000125, A058331, A080856, A000127, A161701-A161704, A161706, A161707, A161708, A161710-A161713, A161715, A006261.
Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), this sequence (g=5), A051890 (g=6), A188377 (g=7). - Jason Kimberley, Oct 30 2011
Cf. A002496 (primes).
Cf. A254858.
Cf. A302612, A302644, A302645, A302646.
Subsequence of A031396.

a002522 = (+ 1) . (^ 2)
a002522_list = scanl (+) 1 [1,3..]
-- Reinhard Zumkeller, Apr 06 2012

[n^2 + 1: n in [0..50]]; // Vincenzo Librandi, May 01 2011

A002522 := proc(n)
        numtheory[cyclotomic](4,n) ;
end proc:
seq(A002522(n),n=0..20) ; # R. J. Mathar, Feb 07 2014

Table[n^2 + 1, {n, 0, 50}]; (* Vladimir Joseph Stephan Orlovsky, Dec 15 2008 *)

A002522(n):=n^2+1$ makelist(A002522(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

a(n)=n^2+1 \\ Charles R Greathouse IV, Jun 10 2011

O.g.f.: (1-x+2*x^2)/((1-x)^3). - Eric Werley, Jun 27 2011
Sequences of the form a(n) = n^2 + K with offset 0 have o.g.f. (K - 2*K*x + K*x^2 + x + x^2)/(1-x)^3 and recurrence a(n) = 3*a(n-1) - 3*a(n-2) + a*(n-3). - R. J. Mathar, Apr 28 2008
For n > 0: a(n-1) = A143053(A000290(n)) - 1. - Reinhard Zumkeller, Jul 20 2008
A143053(a(n)) = A000290(n+1). - Reinhard Zumkeller, Jul 20 2008
a(n)*a(n-2) = (n-1)^4 + 4. - Reinhard Zumkeller, Feb 12 2009
a(n) = A156798(n)/A087475(n). - Reinhard Zumkeller, Feb 16 2009
From Reinhard Zumkeller, Mar 08 2010: (Start)
a(n) = A170949(A002061(n+1));
A170949(a(n)) = A132411(n+1);
A170950(a(n)) = A002061(n+1). (End)
For n > 1, a(n)^2 + (a(n) + 1)^2 + ... + (a(n) + n - 2)^2 + (a(n) + n - 1 + a(n) + n)^2 = (n+1) *(6*n^4 + 18*n^3 + 26*n^2 + 19*n + 6) / 6 = (a(n) + n)^2 + ... + (a(n) + 2*n)^2. - Charlie Marion, Jan 10 2011
From Eric Werley, Jun 27 2011: (Start)
a(n) = 2*a(n-1) - a(n-2) + 2.
a(n) = a(n-1) + 2*n - 1. (End)
a(n) = (n-1)^2 + 2(n-1) + 2 = 122 read in base n-1 (for n > 3). - Jason Kimberley, Oct 20 2011
a(n)*a(n+1) = a(n*(n+1) + 1) so a(1)*a(2) = a(3). More generally, a(n)*a(n+k) = a(n*(n+k) + 1) + k^2 - 1. - Jon Perry, Aug 01 2012
a(n) = (n!)^2* [x^n] BesselI(0, 2*sqrt(x))*(1+x). - Peter Luschny, Aug 25 2012
a(n) = A070216(n,1) for n > 0. - Reinhard Zumkeller, Nov 11 2012
E.g.f.: exp(x)*(1 + x + x^2). - Geoffrey Critzer, Aug 30 2013
a(n) = A254858(n-2,3) for n > 2. - Reinhard Zumkeller, Feb 09 2015
Sum_{n>=0} (-1)^n / a(n) = (1+Pi/sinh(Pi))/2 = 0.636014527491... = A367976 . - Vaclav Kotesovec, Feb 14 2015
Sum_{n>=0} 1/a(n) = (1 + Pi*coth(Pi))/2 = 2.076674... = A113319. - Vaclav Kotesovec, Apr 10 2016
4*a(n) = A001105(n-1) + A001105(n+1). - Bruno Berselli, Jul 03 2017
From Amiram Eldar, Jan 20 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi)*sinh(sqrt(2)*Pi).
Product_{n>=1} (1 - 1/a(n)) = Pi*csch(Pi). (End)
Sum_{n>=0} a(n)/n! = 3*e. - Davide Rotondo, Feb 16 2025

Partially edited by Joerg Arndt, Mar 11 2010

A049532 Numbers k such that k^2 + 1 is not squarefree.

Original entry on oeis.org

7, 18, 32, 38, 41, 43, 57, 68, 70, 82, 93, 99, 107, 117, 118, 132, 143, 157, 168, 182, 193, 207, 218, 232, 239, 243, 251, 257, 268, 282, 293, 307, 318, 327, 332, 343, 357, 368, 378, 382, 393, 407, 408, 418, 432, 437, 443, 457, 468, 482, 493, 500, 507, 515

Offset: 1

Labos Elemer

nonn

The sequence is infinite. For instance, it contains all numbers of the form 7 + 25m. - Emmanuel Vantieghem, Oct 25 2016
More generally, the sequence contains all numbers of the form a(n) + (a(n)^2 + 1) * m for even a(n) and a(n) + (a(n)^2 + 1) * m / 2 for odd a(n). - David A. Corneth, Oct 25 2016
The asymptotic density of this sequence is 1 - A335963 = 0.1051587754... - Amiram Eldar, Jul 08 2020

			a(1) = 7 because 7^2 + 1 = 49 + 1 = 50 is divisible by 25, a square.

R. J. Mathar, Table of n, a(n) for n = 1..7999

Cf. A002522, A059592, A124809, A335963.

[n: n in [1..6*10^2]| not IsSquarefree(n^2+1)]; // Bruno Berselli, Oct 15 2012

n=1;Reap[Do[While[SquareFreeQ[n^2+1],n++];Sow[n];n++,{c,10000}]][[2,1]] (* Zak Seidov, Feb 24 2011 *)

for(n=1,1e4,if(!issquarefree(n^2+1),print1(n", "))) \\ Charles R Greathouse IV, Feb 24 2011

A059592(a(n)) > 1; A124809(n) = a(n)^2 + 1. - Reinhard Zumkeller, Nov 08 2006

Definition rewritten by Bruno Berselli, Oct 15 2012

Mathematica updated by Jean-François Alcover, Jun 19 2013

A049533 Numbers k such that k^2+1 is squarefree.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 39, 40, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Labos Elemer

nonn

Estermann proved that a(n) ~ kn with k = 1.117...; more precisely, there are cx + O(x^(2/3) log x) terms up to x, where c = 1/k = Product (1 - 2/p^2) where the product is over primes p which are 1 mod 4. Heath-Brown improves the error term to O(x^(7/12) log x). - Charles R Greathouse IV, Oct 16 2017, corrected by Amiram Eldar, Jul 08 2020

There are 89489 terms up to 10^5, 894856 terms up to 10^6, 8948417 up to 10^7, 89484102 up to 10^8, and 894841314 up to 10^9. - Charles R Greathouse IV, Nov 26 2017, corrected and extended by Amiram Eldar, Jul 08 2020

			10 is a member because 10^2 + 1 = 100 + 1 = 101 is squarefree.
Reasons why certain numbers are excluded: 7^2+1 = 2*5^2, 18^2+1 = 13*5^2, 32^2+1 = 41*5^2, 38^2+1 = 5*17^2, 41^2+1 = 2*29^2, 43^2+1 = 74*5^2, 57^2+1 = 130*5^2, 82^2+1 = 269*5^2. - Neven Juric, Oct 06 2008

Michael De Vlieger, Table of n, a(n) for n = 1..10000
T. Estermann, Einige Sätze über quadratfreie Zahlen, Math. Ann. 105 (1931), pp. 653-662.
D. R. Heath-Brown, Square-free values of n^2 + 1, Acta Arithmetica 155:1 (2012), pp. 1-13; arXiv:1010.6217 [math.NT], 2010-2012.

Complement of A049532.

Cf. A059592, A069987, A335963.

[ n: n in [1..100] | IsSquarefree(n^2+1) ]; // Vincenzo Librandi, Dec 25 2010

Select[Range@ 80, SquareFreeQ[#^2 + 1] &] (* Michael De Vlieger, Aug 09 2017 *)

isok(n) = issquarefree(n^2+1); \\ Michel Marcus, Feb 09 2016

Numbers k such that A059592(k) = 1. - Reinhard Zumkeller, Nov 08 2006

A059321 Smallest number m such that m^2+1 is divisible by A002144(n)^2 (= squares of primes congruent to 1 mod 4).

Original entry on oeis.org

7, 70, 38, 41, 117, 378, 500, 682, 776, 3861, 4052, 515, 5744, 1710, 6613, 1744, 11018, 13241, 3458, 5099, 1393, 16610, 26884, 15006, 2072, 13637, 31361, 4443, 26508, 7850, 37520, 31152, 39922, 37107, 6072, 4005, 32491, 4030, 43211, 12238

Offset: 1

Marc LeBrun, Jan 26 2001

a(2) = 70 since A002144(2)=13, 70^2+1 = 4091 = 13^2 * 29 and for no k<70 does 13^2 divide k^2+1. Related to period-1 continued fractions.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Klaus Brockhaus)

Cf. A002144, A059591, A059592.

from itertools import islice
from sympy import nextprime, sqrt_mod_iter
def A059321_gen(): # generator of terms
    p = 1
    while (p:=nextprime(p)):
        if p&3==1:
            yield min(sqrt_mod_iter(-1,p**2))
A059321_list = list(islice(A059321_gen(),20)) # Chai Wah Wu, May 04 2024

A059591 Squarefree part of n^2+1.

Original entry on oeis.org

1, 2, 5, 10, 17, 26, 37, 2, 65, 82, 101, 122, 145, 170, 197, 226, 257, 290, 13, 362, 401, 442, 485, 530, 577, 626, 677, 730, 785, 842, 901, 962, 41, 1090, 1157, 1226, 1297, 1370, 5, 1522, 1601, 2, 1765, 74, 1937, 2026, 2117, 2210, 2305, 2402, 2501, 2602

Offset: 0

Marc LeBrun, Jan 25 2001

Related to period-1 continued fractions [z,z,z,...].

			a(7)=2 since 7^2+1 = 50 = 25*2 = (5^2)*2.

Enrique Pérez Herrero, Table of n, a(n) for n = 0..5000

Cf. A007913, A059592, A000188.

A:= proc(n)
local F;
F:= select(t -> t[2]::odd, ifactors(n^2+1)[2]);
mul(t[1],t=F)
end proc:
map(A, [$0..100]); # Robert Israel, Jun 18 2015

Table[Times @@ Flatten[Table[#1, {#2}] & @@@ Select[FactorInteger[n^2 + 1], OddQ@ Last@ # &]], {n, 120}] (* Michael De Vlieger, Jun 19 2015 *)

a(n) = core(n^2+1); \\ Michel Marcus, Jun 18 2015

a(n) = A007913(n^2+1).

a(n)*A059592(n)^2 = n^2+1.

A124808 Number of numbers k <= n such that k^2 + 1 is squarefree.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 35, 36, 37, 37, 38, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 61, 62, 62, 63, 64

Offset: 0

Reinhard Zumkeller, Nov 08 2006

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A008683, A049532, A059592, A335963.

ListTools:-PartialSums([seq(numtheory:-mobius(k^2+1)^2, k=0..100)]); # Robert Israel, Jul 15 2015

Accumulate[Table[If[SquareFreeQ[k^2+1],1,0],{k,0,80}]] (* Harvey P. Dale, Mar 04 2014 *)
Table[Sum[MoebiusMu[k^2 + 1]^2, {k, 0, n}], {n, 0, 100}] (* Wesley Ivan Hurt, Jul 15 2015 *)

a(n)={my(k,r=0);for(k=0,n,if(issquarefree(k^2+1),r++));return(r);}
main(size)=my(n);vector(size,n,a(n-1)) /* Anders Hellström, Jul 15 2015 */

a(0) = 1, a(n) = a(n-1) + 0^(A059592(n) - 1).
a(n) = Sum_{k=0..n} mu(k^2+1)^2, where mu(n) is the Mobius function (A008683). - Wesley Ivan Hurt, Jul 15 2015
a(n) ~ c*n where c = Product_{p prime, p == 1 (mod 4)} (1 - 2/p^2) = 0.894841... (A335963). - Amiram Eldar, Feb 23 2021

A124897 a(n) = mu(n^2 + 1), mu = A008683.

Original entry on oeis.org

-1, -1, 1, -1, 1, -1, 0, 1, 1, -1, 1, 1, -1, -1, 1, -1, -1, 0, 1, -1, -1, 1, -1, -1, 1, -1, -1, 1, 1, 1, -1, 0, -1, 1, 1, -1, -1, 0, 1, -1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, 0, 1, 1, 1, 1, 1, -1, 1, 1, -1, -1, 0, 1, 0, 1, -1, 1, -1, -1, 1, -1, 1, 1, 1, -1, 0, 1, -1, 1, 1, -1, 1, -1, -1, -1, 1, 0, -1, 1, 1, -1, -1, 0, 1, 1, 1, -1, 1, -1, 1

Offset: 1

Reinhard Zumkeller, Nov 12 2006

sign

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

Cf. A002522, A008683, A059592, A124895.

MoebiusMu[Range[107]^2+1] (* T. D. Noe, Nov 27 2006 *)

a(n) = moebius(n^2+1); \\ Michel Marcus, Jul 26 2022

a(n) = A008683(A002522(n)) = A124895(n,1);

a(n) = 0 iff A059592(n) > 1;

Corrected by T. D. Noe, Nov 27 2006

A282092 Numbers m such that there exists at least one integer k < m such that m^2+1 and k^2+1 have the same prime factors.

Original entry on oeis.org

7, 18, 117, 239, 378, 843, 2207, 2943, 4443, 4662, 6072, 8307, 8708, 9872, 31561, 103682, 271443, 853932, 1021693, 3539232, 3699356, 6349657, 6907607, 7042807, 7249325, 9335094, 12623932, 12752043, 12813848, 22211431, 33385282, 42483057, 52374157, 105026693

Offset: 1

Michel Lagneau, Feb 06 2017

nonn

For the pairs (m, k), is k always unique?

The pairs (m, k) are (7, 3), (18, 8), (117, 43), (239, 5), (378, 132), (843, 377), (2207, 987), (2943, 73), (4443, 53), (4662, 1568), (6072, 5118), (8307, 743), (8708, 2112), (9872, 2738), ...

			7 is in the sequence because of the pair (m, k) = (7, 3), 7^2+1 = 2*5^2 and 3^2+1 = 2*5 with the same prime factors 2 and 5.

Subsequence of A049532 (numbers n such that n^2 + 1 is not squarefree).

Cf. A002522, A059591, A059592, A124809.

Select[Range@ 5000, Function[m, Total@ Boole@ Table[Function[w, And[SameQ[First@ w, #], SameQ[Last@ w, #]] &@ Union@ Flatten@ w]@ Map[FactorInteger[#][[All, 1]] &, {m^2 + 1, k^2 + 1}], {k, m - 1}] > 0]] (* Michael De Vlieger, Feb 07 2017 *)

isok(n)=ok = 0; vn = factor(n^2+1)[,1]; for (k=1, n-1, if (factor(k^2+1)[,1] == vn, ok = 1; break);); ok; \\ Michel Marcus, Feb 09 2017

squeeze(f)=factorback(f)\2
list(lim)=my(v=List(),m=Map(),t); for(n=1,lim, t=squeeze(factor(n^2+1)[,1]); if(mapisdefined(m,t), listput(v,n), mapput(m,t,0))); Vec(v) \\ Charles R Greathouse IV, Feb 12 2017

use ntheory qw(:all);
for (my ($m, %t) = 1 ; ; ++$m) {
my $k = vecprod(map{$_->[0]}factor_exp($m**2+1));
push @{$t{$k}}, $m;
if (@{$t{$k}} >= 2) {
print'('.join(', ',reverse(@{$t{$k}})).")\n";
}
} # Daniel Suteu, Feb 08 2017

a(15)-a(29) from Daniel Suteu, Feb 08 2017
a(30) from Daniel Suteu, Feb 10 2017
a(31)-a(34) from Joerg Arndt, Feb 11 2017

cp's OEIS Frontend

A002522 a(n) = n^2 + 1.

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A049532 Numbers k such that k^2 + 1 is not squarefree.

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A049533 Numbers k such that k^2+1 is squarefree.

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A059321 Smallest number m such that m^2+1 is divisible by A002144(n)^2 (= squares of primes congruent to 1 mod 4).

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A059591 Squarefree part of n^2+1.

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A124808 Number of numbers k <= n such that k^2 + 1 is squarefree.

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A124897 a(n) = mu(n^2 + 1), mu = A008683.