A069987 -id:A069987 - cp's OEIS frontend

A059100 a(n) = n^2 + 2.

2, 3, 6, 11, 18, 27, 38, 51, 66, 83, 102, 123, 146, 171, 198, 227, 258, 291, 326, 363, 402, 443, 486, 531, 578, 627, 678, 731, 786, 843, 902, 963, 1026, 1091, 1158, 1227, 1298, 1371, 1446, 1523, 1602, 1683, 1766, 1851, 1938, 2027, 2118, 2211, 2306, 2403, 2502, 2603

Offset: 0

Henry Bottomley, Feb 13 2001

Let s(n) = Sum_{k>=1} 1/n^(2^k). Then I conjecture that the maximum element in the continued fraction for s(n) is n^2 + 2. - Benoit Cloitre, Aug 15 2002
Binomial transformation yields A081908, with A081908(0)=1 dropped. - R. J. Mathar, Oct 05 2008
1/a(n) = R(n)/r with R(n) the n-th radius of the Pappus chain of the symmetric arbelos with semicircle radii r, r1 = r/2 = r2. See the MathWorld link for Pappus chain (there are two of them, a left and a right one. In this case these two chains are congruent). - Wolfdieter Lang, Mar 01 2013
a(n) is the number of election results for an election with n+2 candidates, say C1, C2, ..., and C(n+2), and with only two voters (each casting a single vote) that have C1 and C2 receiving the same number of votes. See link below. - Dennis P. Walsh, May 08 2013
This sequence gives the set of values such that for sequences b(k+1) = a(n)*b(k) - b(k-1), with initial values b(0) = 2, b(1) = a(n), all such sequences are invariant under this transformation: b(k) = (b(j+k) + b(j-k))/b(j), except where b(j) = 0, for all integer values of j and k, including negative values. Examples are: at n=0, b(k) = 2 for all k; at n=1, b(k) = A005248; at n=2, b(k) = 2*A001541; at n=3, b(k)= A057076; at n=4, b(k) = 2*A023039. This b(k) family are also the transformation results for all related b'(k) (i.e., those with different initial values) including non-integer values. Further, these b(k) are also the bisections of the transformations of sequences of the form G(k+1) = n * G(k) + G(k-1), and those bisections are invariant for all initial values of g(0) and g(1), including non-integer values. For n = 1 this g(k) family includes Fibonacci and Lucas, where the invariant bisection is b(k) = A005248. The applicable bisection for this transformation of g(k) is for the odd values of k, and applies for all n. Also see A000032 for a related family of sequences. - Richard R. Forberg, Nov 22 2014
Also the number of maximum matchings in the n-gear graph. - Eric W. Weisstein, Dec 31 2017
Also the Wiener index of the n-dipyramidal graph. - Eric W. Weisstein, Jun 14 2018
Numbers of the form n^2+2 have no factors that are congruent to 7 (mod 8). - Gordon E. Michaels, Sep 12 2019
For n >= 1, the continued fraction expansion of sqrt(a(n)) is [n; {n, 2n}]. - Magus K. Chu, Sep 10 2022

			For n = 2, a(2) = 6 since there are 6 election results in a 4-candidate, 2-voter election that have candidates c1 and c2 tied. Letting <i,j> denote voter 1 voting for candidate i and voter 2 voting for candidate j, the six election results are <1,2>, <2,1>, <3,3>, <3,4>, <4,3>, and <4,4>. - _Dennis P. Walsh_, May 08 2013

Harry J. Smith, Table of n, a(n) for n = 0..1000
Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 52, 56.
Hesam Dashti, A New Upper Bound on the Length of Shortest Permutation Strings; An Algorithm for Generating Permutation Strings, arXiv:1009.5053 [math.CO], 2010. - _Jonathan Vos Post_, Sep 28 2010
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Alexander Soifer, Coffee Hour and the Conway-Soifer Cover-Up, In: How Does One Cut a Triangle? (2009), pp. 147-156. See also here
Dennis P. Walsh, Notes on a tied election.
Eric Weisstein's World of Mathematics, Dipyramidal Graph.
Eric Weisstein's World of Mathematics, Gear Graph.
Eric Weisstein's World of Mathematics, Matching.
Eric Weisstein's World of Mathematics, Maximum Independent Edge Set.
Eric Weisstein's World of Mathematics, Near-Square Prime.
Eric Weisstein's World of Mathematics, Pappus Chain.
Eric Weisstein's World of Mathematics, Wiener Index.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000032, A000196, A000290, A001541, A002522, A005248, A008865, A056105, A056109, A056899, A057076, A069987, A081908, A114964, A156798, A166464.
Apart from initial terms, same as A010000.
2nd row/column of A295707.

a059100 = (+ 2) . (^ 2)
a059100_list = scanl (+) (2) [1, 3 ..]
-- Reinhard Zumkeller, Feb 09 2015

with(combinat, fibonacci):seq(fibonacci(3, i)+1, i=0..49); # Zerinvary Lajos, Mar 20 2008

Table[n^2 + 2, {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, Dec 15 2008 *)
LinearRecurrence[{3, -3, 1}, {2, 3, 6}, 50] (* Vincenzo Librandi, Feb 15 2012 *)
Range[0, 20]^2 + 2 (* Eric W. Weisstein, Dec 31 2017 *)
CoefficientList[Series[(-2 + 3 x - 3 x^2)/(-1 + x)^3, {x, 0, 20}], x] (* Eric W. Weisstein, Dec 31 2017 *)

a(n) = { n^2+2 } \\ Harry J. Smith, Jun 24 2009

[lucas_number1(3,n,-2) for n in range(0, 50)] # Zerinvary Lajos, May 16 2009

G.f.: (2 - 3*x + 3*x^2)/(1 - x)^3. - R. J. Mathar, Oct 05 2008
a(n) = ((n - 2)^2 + 2*(n + 1)^2)/3. - Reinhard Zumkeller, Feb 13 2009
a(n) = A000196(A156798(n) - A000290(n)). - Reinhard Zumkeller, Feb 16 2009
a(n) = 2*n + a(n-1) - 1 with a(0) = 2. - Vincenzo Librandi, Aug 07 2010
a(n+3) = (A166464(n+5) - A166464(n))/20. - Paul Curtz, Nov 07 2012
From Paul Curtz, Nov 07 2012: (Start)
a(3*n) mod 9 = 2.
a(3*n+1) = 3*A056109(n).
a(3*n+2) = 3*A056105(n+1). (End)
Sum_{n >= 1} 1/a(n) = Pi * coth(sqrt(2)*Pi) / 2^(3/2) - 1/4. - Vaclav Kotesovec, May 01 2018
From Amiram Eldar, Jan 29 2021: (Start)
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(2)*Pi*(csch(sqrt(2)*Pi)))/4.
Product_{n>=0} (1 + 1/a(n)) = sqrt(3/2)*csch(sqrt(2)*Pi)*sinh(sqrt(3)*Pi).
Product_{n>=0} (1 - 1/a(n)) = csch(sqrt(2)*Pi)*sinh(Pi)/sqrt(2). (End)
E.g.f.: exp(x)*(2 + x + x^2). - Stefano Spezia, Aug 07 2024

A144255 Semiprimes of the form k^2+1.

Original entry on oeis.org

10, 26, 65, 82, 122, 145, 226, 362, 485, 626, 785, 842, 901, 1157, 1226, 1522, 1765, 1937, 2026, 2117, 2305, 2402, 2501, 2602, 2705, 3365, 3482, 3601, 3722, 3845, 4097, 4226, 4762, 5042, 5777, 6085, 6242, 6401, 7226, 7397, 7745, 8465, 9026, 9217, 10001, 10202

Offset: 1

T. D. Noe, Sep 16 2008

Iwaniec proves that there are an infinite number of semiprimes or primes of the form n^2+1. Because n^2+1 is not a square for n>0, all such semiprimes have two distinct prime factors.

Moreover, this implies that one prime factor p of n^2+1 is strictly smaller than n, and therefore also divisor of (the usually much smaller) m^2+1, where m = n % p (binary "mod" operation). - M. F. Hasler, Mar 11 2012

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Henryk Iwaniec, Almost-primes represented by quadratic polynomials, Inventiones Mathematicae 47 (2) (1978), pp. 171-188.

Cf. A001358, A085722, A069987, A193432.

Subsequence of A134406.

IsSemiprime:= func; [s: n in [1..100] | IsSemiprime(s) where s is n^2 + 1]; // Vincenzo Librandi, Sep 22 2012

Select[Table[n^2  + 1, {n, 100}], PrimeOmega[#] == 2&] (* Vincenzo Librandi, Sep 22 2012 *)

select(n->bigomega(n)==2,vector(500,n,n^2+1)) \\ Zak Seidov Feb 24 2011

from sympy import primeomega
from itertools import count, takewhile
def aupto(limit):
    form = takewhile(lambda x: x <= limit, (k**2+1 for k in count(1)))
    return [number for number in form if primeomega(number)==2]
print(aupto(10202)) # Michael S. Branicky, Oct 26 2021

a(n) = A085722(n)^2 + 1.

Equals { n^2+1 | A193432(n)=2 }. - M. F. Hasler, Mar 11 2012

A335963 Decimal expansion of Product_{p prime, p == 1 (mod 4)} (1 - 2/p^2).

Original entry on oeis.org

8, 9, 4, 8, 4, 1, 2, 2, 4, 5, 6, 2, 4, 8, 8, 1, 7, 0, 7, 2, 5, 6, 6, 1, 5, 0, 6, 9, 0, 8, 4, 3, 7, 3, 2, 1, 9, 8, 7, 5, 4, 7, 8, 0, 8, 9, 2, 0, 7, 1, 8, 9, 7, 2, 6, 0, 1, 7, 9, 9, 4, 2, 7, 6, 1, 6, 5, 6, 3, 8, 9, 2, 2, 1, 2, 0, 9, 1, 5, 5, 0, 2, 8, 8, 5, 9, 4, 2, 9, 1, 0, 5, 3, 9, 5, 8, 9, 1, 0, 8, 0, 0, 3, 3, 2, 2

Offset: 0

Amiram Eldar, Jul 01 2020

The asymptotic density of the numbers k such that k^2+1 is squarefree (A049533) (Estermann, 1931).
The constant c in Sum_{k=0..n} phi(k^2 + 1) = A333170(n) ~ (1/4)*c*n^3 (Finch, 2018).
The constant c in Sum_{k=0..n} phi(k^2 + 1)/(k^2 + 1) = (3/4)*c*n + O(log(n)^2) (Postnikov, 1988).

			0.89484122456248817072566150690843732198754780892071...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 101.
Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 166.
A. G. Postnikov, Introduction to Analytic Number Theory, Amer. Math. Soc., 1988, pp. 192-195.

Theodor Estermann, Einige Sätze über quadratfreie Zahlen, Mathematische Annalen, Vol. 105 (1931), pp. 653-662, alternative link.
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, p. 14.
D. R. Heath-Brown, Square-free values of n^2 + 1, Acta Arithmetica, Vol. 155, No. 1 (2012), pp. 1-13.
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo functions for small moduli, arXiv:1008.2547 [math.NT], variable F(m=4,n=1,s=2), p. 38.
Wolfgang Schwarz, Über die Summe Sigma_{n <= x} phi(f)(n) und verwandte Probleme, Monatshefte für Mathematik, Vol. 66, No. 1 (1962), pp. 43-54, alternative link.
Radoslav Tsvetkov, On the distribution of k-free numbers and r-tuples of k-free numbers. A survey, Notes on Number Theory and Discrete Mathematics, Vol. 25, No. 3 (2019), pp. 207-222.

Cf. A002144, A049533, A065474, A069987, A088539, A333169, A333170, A340617.

Digits := 150;
with(NumberTheory);
DirichletBeta := proc(s) (Zeta(0, s, 1/4) - Zeta(0, s, 3/4))/4^s; end proc;
alfa := proc(s) DirichletBeta(s)*Zeta(s)/((1 + 1/2^s)*Zeta(2*s)); end proc;
beta := proc(s) (1 - 1/2^s)*Zeta(s)/DirichletBeta(s); end proc;
pzetamod41 := proc(s, terms) 1/2*Sum(Moebius(2*j + 1)*log(alfa((2*j + 1)*s))/(2*j + 1), j = 0..terms); end proc;
evalf(exp(-Sum(2^t*pzetamod41(2*t, 50)/t, t = 1..200))); # Vaclav Kotesovec, Jan 13 2021

f[p_] := If[Mod[p, 4] == 1, 1 - 2/p^2, 1]; RealDigits[N[Product[f[Prime[i]], {i, 1, 10^6}], 10], 10, 8][[1]] (* for calculating only the first few terms *)
(* -------------------------------------------------------------------------- *)
S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);
P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]] * S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];
Z2[m_, n_, s_] := (w = 1; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = 2^w * P[m, n, s*w]/w; sumz = sumz + difz; w++]; Exp[-sumz]);
$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Z2[4, 1, 2], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *)

f(lim,poly=1-'x-'x^2/2)=prodeulerrat(subst(poly,'x,1/'x^2))*prodeuler(p=2,lim, my(pm2=1./p^2); if(p%4==1,1.-2*pm2,1.)/subst(poly,'x,pm2)) \\ Gets 14 digits at lim=1e9; Charles R Greathouse IV, Aug 10 2022

Equals 2*A065474/A340617.

More digits (from the paper by R. J. Mathar) added by Jon E. Schoenfield, Jan 12 2021

More digits from Vaclav Kotesovec, Jan 13 2021

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

A124809 Numbers of the form (square + 1) that are not squarefree.

Original entry on oeis.org

50, 325, 1025, 1445, 1682, 1850, 3250, 4625, 4901, 6725, 8650, 9802, 11450, 13690, 13925, 17425, 20450, 24650, 28225, 33125, 37250, 42850, 47525, 53825, 57122, 59050, 63002, 66050, 71825, 79525, 85850, 94250, 101125, 106930, 110225, 117650

Offset: 1

Reinhard Zumkeller, Nov 08 2006

nonn

The sequence is infinite (see comment in A049532). - Emmanuel Vantieghem, Oct 25 2016

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A002522, A013929, A049532, A069987.

[n^2+1: n in [1..500]| not IsSquarefree(n^2+1)]; // Vincenzo Librandi, Oct 26 2016

Select[Range[400]^2+1,!SquareFreeQ[#]&] (* Harvey P. Dale, Sep 15 2016 *)

is(n)=issquare(n-1) && !issquarefree(n) \\ Charles R Greathouse IV, Nov 05 2017

a(n) = A049532(n)^2 + 1.

A335962 Numbers k such that k^2 + 1 and k^2 + 2 are both squarefree.

Original entry on oeis.org

1, 2, 3, 6, 8, 9, 10, 11, 12, 15, 16, 17, 20, 21, 25, 26, 27, 28, 29, 30, 33, 34, 35, 36, 37, 39, 42, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 56, 60, 61, 62, 64, 65, 66, 69, 72, 73, 74, 75, 78, 79, 80, 81, 83, 84, 87, 88, 89, 90, 91, 92, 96, 97, 98, 100, 101

Offset: 1

Amiram Eldar, Jul 01 2020

nonn

Dimitrov (2020) proved that this sequence is infinite and has an asymptotic density Product_{p prime > 2} (1 - ((-1/p) + (-2/p) + 2)/p^2) = 0.67187..., where (a/p) is the Legendre symbol.

			1 is a term since 1^2 + 1 = 2 and 1^1 + 2 = 3 are both squarefree.

Amiram Eldar, Table of n, a(n) for n = 1..10000
S. I. Dimitrov, Pairs of square-free values of the type n^2+1, n^2+2, arXiv:2004.09975 [math.NT], 2020.

Subsequence of A049533.

Cf. A005117, A007674, A069987.

Select[Range[100], And @@ SquareFreeQ /@ (#^2 + {1, 2}) &]

A248742 Numbers of the form x^2+1 with at most two prime factors.

Original entry on oeis.org

2, 5, 10, 17, 26, 37, 65, 82, 101, 122, 145, 197, 226, 257, 362, 401, 485, 577, 626, 677, 785, 842, 901, 1157, 1226, 1297, 1522, 1601, 1765, 1937, 2026, 2117, 2305, 2402, 2501, 2602, 2705, 2917, 3137, 3365, 3482, 3601, 3722, 3845, 4097, 4226, 4357, 4762

Offset: 1

R. J. Mathar, Oct 13 2014

nonn

Prime factors are counted with multiplicity, as in A144255.

Iwaniec shows that the sequence is infinite.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
H. Iwaniec, Almost-primes represented by quadratic polynomials, Inventiones Mathematicae 47:2 (1978), pp. 171-188.
Vishaal Kapoor, Almost-primes represented by quadratic polynomials, MS thesis (2006). arXiv:1910.02885 [math.NT]
Robert J. Lemke Oliver, Almost-primes represented by quadratic polynomials, Acta Arithm. 151 (2012) 241. DOI: 10.4064/aa151-3-2
Wikipedia, Landau's problems

Cf. A002496, A069987.

is(n)=issquare(n-1) && bigomega(n)<3 \\ Charles R Greathouse IV, Feb 05 2017

A002496 UNION A144255.

A268641 Squarefree numbers k such that k^2 + 1 and k^2 - 1 are also squarefree.

Original entry on oeis.org

2, 6, 14, 22, 30, 34, 42, 58, 66, 78, 86, 94, 102, 106, 110, 114, 130, 138, 142, 158, 166, 178, 186, 194, 202, 210, 214, 222, 230, 238, 254, 258, 266, 286, 302, 310, 322, 330, 346, 354, 358, 366, 390, 394, 398, 402, 410, 430, 434, 438, 446, 454, 462, 466, 470, 498

Offset: 1

K. D. Bajpai, Feb 09 2016

nonn

All the listed terms are even squarefree numbers.

Subsequence of A039956.

			a(2) = 6 = 2 * 3: 6^2 + 1 = 37 = 1 * 37; 6^2 - 1 = 35 = 5 * 7; 6, 37, 35 are all squarefree.

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

Cf. A005117, A007675, A039956, A049532, A049533, A069987, A173186.

[n : n in [1..1000]  |  IsSquarefree(n) and IsSquarefree(n^2+1) and IsSquarefree(n^2-1) ];

select(n -> andmap(issqrfree, [n, n^2+1, n^2-1]), [seq(n, n=2.. 10^3)]);

Select[Range[1000], SquareFreeQ[#] && SquareFreeQ[#^2 + 1] && SquareFreeQ[#^2 - 1] &]

for(n=2, 1000, issquarefree(n) & issquarefree(n^2 + 1) & issquarefree(n^2 - 1) & print1(n,", "))

cp's OEIS Frontend

A059100 a(n) = n^2 + 2.

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A144255 Semiprimes of the form k^2+1.

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A335963 Decimal expansion of Product_{p prime, p == 1 (mod 4)} (1 - 2/p^2).

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

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A124809 Numbers of the form (square + 1) that are not squarefree.

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A335962 Numbers k such that k^2 + 1 and k^2 + 2 are both squarefree.

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A248742 Numbers of the form x^2+1 with at most two prime factors.