A049533 -id:A049533 - cp's OEIS frontend

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

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

A069987 Squarefree numbers of form k^2 + 1.

Original entry on oeis.org

2, 5, 10, 17, 26, 37, 65, 82, 101, 122, 145, 170, 197, 226, 257, 290, 362, 401, 442, 485, 530, 577, 626, 677, 730, 785, 842, 901, 962, 1090, 1157, 1226, 1297, 1370, 1522, 1601, 1765, 1937, 2026, 2117, 2210, 2305, 2402, 2501, 2602, 2705, 2810, 2917, 3026

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 01 2002

nonn

Heath-Brown (following Estermann) shows that, for any e > 0, there are k sqrt(x) + O(x^{7/24 + e}) members of this sequence up to x, for k = Product(1 - 2/p^2) = 0.8948412245... (A335963) where the product is over primes p = 1 mod 4. - Charles R Greathouse IV, Nov 19 2012, corrected by Amiram Eldar, Jul 08 2020

Integers k for which the period of the continued fraction of sqrt(k) is 1. - Michel Marcus, Apr 12 2019

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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, arXiv:1010.6217 [math.NT], 2010-2012.
D. R. Heath-Brown, Square-free values of n^2 + 1, Acta Arithmetica 155 (2012), pp. 1-13.

Cf. A059591, A002496, A124809, A005117, A002522, A335963.

select(numtheory:-issqrfree, [seq(n^2+1, n=1..100)]); # Robert Israel, Feb 09 2016

Select[ Range[10^4], IntegerQ[ Sqrt[ # - 1]] && Union[ Transpose[ FactorInteger[ # ]] [[2]]] [[ -1]] == 1 &]
Select[Range[60]^2+1,SquareFreeQ] (* Harvey P. Dale, Mar 21 2013 *)

for(n=1,100,if(issquarefree(n^2+1),print1(n^2+1,",")))

a(n) = A049533(n)^2 + 1.

Edited and extended by Robert G. Wilson v, Benoit Cloitre and Vladeta Jovovic, May 04 2002

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}) &]

A124895 Triangle read by rows, 1 <= k <= n: T(n,k) = mu(n^2 + k^2) with mu = A008683.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Nov 12 2006

			Triangle begins:
 -1
 -1,  0
  1, -1,  0
 -1,  0,  0,  0
  1, -1,  1, -1,  0
 -1,  0,  0,  0, -1, 0
  0, -1,  1,  1,  1, 1,  0
  1,  0, -1,  0, -1, 0, -1, 0
  1,  1,  0, -1,  1, 0, -1, 1,  0
 -1,  0, -1,  0,  0, 0, -1, 0, -1, 0

Cf. A000007, A008683, A049532, A049533, A070216, A124897.

row[n_] := Table[MoebiusMu[n^2 + k^2], {k, 1, n}]; Array[row, 15] // Flatten (* Amiram Eldar, May 12 2025 *)

row(n) = vector(n, k, moebius(n^2 + k^2)); \\ Amiram Eldar, May 12 2025

T(n,k) = A008683(A070216(n,k)).
T(n,1) = A124897(1); T(A049533(n),1) <> 0; T(A049532(n),1) = 0.
T(n,n) = -A000007(n-1).
A124896(n) = Sum_{k=1..n} abs(T(n,k)), row sums of absolute values.

A225856 Primes p such that p^2 + 1 is squarefree.

Original entry on oeis.org

2, 3, 5, 11, 13, 17, 19, 23, 29, 31, 37, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 151, 163, 167, 173, 179, 181, 191, 197, 199, 211, 223, 227, 229, 233, 241, 263, 269, 271, 277, 281, 283, 311, 313, 317, 331, 337

Offset: 1

Rafael Parra Machio, May 18 2013

nonn

Primes of the sequence A224718 generating squarefree.

			23 is a term since 23^2+1 = 530 = 2*5*53, is squarefree.
43 is not a term since 43^2+1 = 1850 = 2*5^2*7, is not squarefree.

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

Intersection of A000040 and A049533.

Cf. A224718.

Select[Prime[Range[100]], SquareFreeQ[#^2+1]&]

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,", "))

A381136 a(n) is the number of divisors d of n such that tau(d^(1 + n) + n) = 2^omega(d^(1 + n) + n), where tau = A000005 and omega = A001221.

Original entry on oeis.org

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

Offset: 1

Juri-Stepan Gerasimov, Feb 15 2025

nonn

Cf. A000005, A001221, A049533, A381138.

[#[d: d in Divisors(n) | #Divisors(d^(1+n)+n) eq 2^#PrimeDivisors(d^(1+n)+n)]: n in [1..50]];

Table[Length[Select[Divisors[n],DivisorSigma[0,#^(1+n)+n]==2^PrimeNu[#^(1+n)+n]&]],{n,45}] (* James C. McMahon, Mar 05 2025 *)

a(n) = sumdiv(n, d, my(f=factor(d^(1 + n) + n)); numdiv(f) == 2^omega(f)); \\ Michel Marcus, Feb 15 2025

More terms from Jinyuan Wang, Mar 09 2025

A359749 Numbers k such that k and k+1 do not share a common exponent in their prime factorizations.

Original entry on oeis.org

1, 3, 4, 7, 8, 9, 15, 16, 24, 25, 26, 27, 31, 32, 35, 36, 48, 63, 64, 71, 72, 81, 100, 107, 108, 120, 121, 124, 125, 127, 128, 143, 144, 168, 169, 195, 196, 199, 200, 215, 216, 224, 225, 242, 243, 255, 256, 287, 289, 323, 342, 361, 391, 392, 399, 400, 431, 432, 440

Offset: 1

Amiram Eldar, Jan 13 2023

nonn

Either k or k+1 is a powerful number (A001694). Except for k=8, are there terms k such that both k and k+1 are powerful (i.e., terms that are also in A060355)? None of the terms A060355(n) for n = 2..39 is in this sequence.

A002496(k)-1, A078324(k)-1, A078325(k)-1, and A049533(k)^2 are terms for all k >= 1.

			3 is a term since 3 has the exponent 1 in its prime factorization, and 3 + 1 = 4 = 2^2 has a different exponent in its prime factorization, 2.

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

Subsequences: A000079, A000225 \ {0}, A075408, A359747.

Cf. A001694, A002496, A049533, A060355, A078324, A078325.

q[n_] := UnsameQ @@ Join @@ (Union[FactorInteger[#][[;; , 2]]]& /@ (n + {0, 1})); Join[{1}, Select[Range[400], q]]

lista(nmax) = {my(e1 = [], e2); for(n = 2, nmax, e2 = Set(factor(n)[,2]); if(setintersect(e1, e2) == [], print1(n-1, ", ")); e1 = e2); }

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

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A069987 Squarefree numbers of form k^2 + 1.

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

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A124895 Triangle read by rows, 1 <= k <= n: T(n,k) = mu(n^2 + k^2) with mu = A008683.

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A225856 Primes p such that p^2 + 1 is squarefree.

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

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A381136 a(n) is the number of divisors d of n such that tau(d^(1 + n) + n) = 2^omega(d^(1 + n) + n), where tau = A000005 and omega = A001221.

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