A074661 -id:A074661 - cp's OEIS frontend

A056594 Period 4: repeat [1,0,-1,0]; expansion of 1/(1 + x^2).

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

Offset: 0

Wolfdieter Lang, Aug 04 2000

G.f. is inverse of cyclotomic(4,x). Unsigned: A000035(n+1).
Real part of i^n and imaginary part of i^(n+1), i=sqrt(-1). - Reinhard Zumkeller, Jul 22 2007
The BINOMIAL transform generates A009116(n); the inverse BINOMIAL transform generates (-1)^n*A009116(n). - R. J. Mathar, Apr 07 2008
a(n-1), n >= 1, is the nontrivial Dirichlet character modulo 4, called Chi_2(4;n) (the trivial one is Chi_1(4;n) given by periodic(1,0) = A000035(n)). See the Apostol reference, p. 139, the k = 4, phi(k) = 2 table. - Wolfdieter Lang, Jun 21 2011
a(n-1), n >= 1, is the character of the Dirichlet beta function. - Daniel Forgues, Sep 15 2012
a(n-1), n >= 1, is also the (strongly) multiplicative function h(n) of Theorem 5.12, p. 150, of the Niven-Zuckerman reference. See the formula section. This function h(n) can be employed to count the integer solutions to n = x^2 + y^2. See A002654 for a comment with the formula. - Wolfdieter Lang, Apr 19 2013
This sequence is duplicated in A101455 but with offset 1. - Gary Detlefs, Oct 04 2013
For n >= 2 this gives the determinant of the bipartite graph with 2*n nodes and the adjacency matrix A(n) with elements A(n;1,2) = 1 = A(n;n,n-1), and for 1 < i < n  A(n;i,i+1) = 1 = A(n;i,i-1), otherwise 0. - Wolfdieter Lang, Jun 25 2023

			With a(n-1) = h(n) of Niven-Zuckerman: a(62) = h(63) = h(3^2*7^1) = (-1)^(2*1)*(-1)^(1*3) = -1 = h(3)^2*h(7) = a(2)^2*a(6) = (-1)^2*(-1) = -1. - _Wolfdieter Lang_, Apr 19 2013

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986.
I. S. Gradstein and I. M. Ryshik, Tables of series, products, and integrals, Volume 1, Verlag Harri Deutsch, 1981.
Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers, New York: John Wiley (1980), p. 150.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 32, equation 32:6:1 at page 300.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry and Nikolaos Pantelidis, On pseudo-involutions, involutions and quasi-involutions in the group of almost Riordan arrays, J Algebr Comb 54, 399-423 (2021).
PeakMath, L-functions and the Langlands program (RH Saga S1E2), YouTube video (2024).
Eric Weisstein's World of Mathematics, Kronecker Symbol.
Wikipedia, Dirichlet beta function.
Wikipedia, Kronecker Symbol.
Index entries for linear recurrences with constant coefficients, signature (0,-1).
Index entries for sequences related to Chebyshev polynomials.
Index to sequences related to inverse of cyclotomic polynomials.

Cf. A049310, A074661, A131852, A002654, A146559 (binomial transform).

&cat[ [1, 0, -1, 0]: n in [0..23] ]; // Bruno Berselli, Feb 08 2011

A056594 := n->(1-irem(n,2))*(-1)^iquo(n,2); # Peter Luschny, Jul 27 2011

CoefficientList[Series[1/(1 + x^2), {x, 0, 50}], x]
a[n_]:= KroneckerSymbol[-4,n+1];Table[a[n],{n,0,93}] (* Thanks to Jean-François Alcover. - Wolfdieter Lang, May 31 2013 *)
CoefficientList[Series[1/Cyclotomic[4, x], {x, 0, 100}], x] (* Vincenzo Librandi, Apr 03 2014 *)

A056594(n) := block(
        [1,0,-1,0][1+mod(n,4)]
)$ /* R. J. Mathar, Mar 19 2012 */

PARI
```
{a(n) = real( I^n )}
```
PARI
```
{a(n) = kronecker(-4, n+1) }
```

def A056594(n): return (1,0,-1,0)[n&3] # Chai Wah Wu, Sep 23 2023

G.f.: 1/(1+x^2).
E.g.f.: cos(x).
a(n) = (1/2)*((-i)^n + i^n), where i = sqrt(-1). - Mitch Harris, Apr 19 2005
a(n) = (1/2)*((-1)^(n+floor(n/2)) + (-1)^floor(n/2)).
Recurrence: a(n)=a(n-4), a(0)=1, a(1)=0, a(2)=-1, a(3)=0.
a(n) = T(n, 0) = A053120(n, 0); T(n, x) Chebyshev polynomials of the first kind. - Wolfdieter Lang, Aug 21 2009
a(n) = S(n, 0) = A049310(n, 0); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind.
Sum_{k>=0} a(k)/(k+1) = Pi/4. - Jaume Oliver Lafont, Mar 30 2010
a(n) = Sum_{k=0..n} A101950(n,k)*(-1)^k. - Philippe Deléham, Feb 10 2012
a(n) = (1/2)*(1 + (-1)^n)*(-1)^(n/2). - Bruno Berselli, Mar 13 2012
a(0) = 1, a(n-1) = 0 if n is even, a(n-1) = Product_{j=1..m} (-1)^(e_j*(p_j-1)/2) if the odd n-1 = p_1^(e_1)*p_2^(e_2)*...*p_m^(e_m) with distinct odd primes p_j, j=1..m. See the function h(n) of Theorem 5.12 of the Niven-Zuckerman reference. - Wolfdieter Lang, Apr 19 2013
a(n) = (-4/(n+1)), n >= 0, where (k/n) is the Kronecker symbol. See the Eric Weisstein and Wikipedia links. Thanks to Wesley Ivan Hurt. - Wolfdieter Lang, May 31 2013
a(n) = R(n,0)/2 with the row polynomials R of A127672. This follows from the product of the zeros of R, and the formula Product_{k=0..n-1} 2*cos((2*k+1)*Pi/(2*n)) = (1 + (-1)^n)*(-1)^(n/2), n >= 1 (see the Gradstein and Ryshik reference, p. 63, 1.396 4., with x = sqrt(-1)). - Wolfdieter Lang, Oct 21 2013
a(n) = Sum_{k=0..n} i^(k*(k+1)), where i=sqrt(-1). - Bruno Berselli, Mar 11 2015
Dirichlet g.f. of a(n) shifted right: L(chi_2(4),s) = beta(s) = (1-2^(-s))*(d.g.f. of A034947), see comments by Lang and Forgues. - Ralf Stephan, Mar 27 2015
a(n) = cos(n*Pi/2). - Ridouane Oudra, Sep 29 2024

A368714 Numbers whose maximal exponent in their prime factorization is even.

Original entry on oeis.org

1, 4, 9, 12, 16, 18, 20, 25, 28, 36, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 75, 76, 80, 81, 84, 90, 92, 98, 99, 100, 112, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 180, 188, 192, 196, 198, 204, 207, 208

Offset: 1

Amiram Eldar, Jan 04 2024

First differs from A240112 at n = 30.
Numbers k such that A051903(k) is even.
The asymptotic density of this sequence is Sum_{k>=2} (-1)^k * (1 - 1/zeta(k)) = 0.27591672059822700769... .

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

Cf. A051903, A240112.
Subsequences: A000290, A000302, A000583, A001014, A001016, A001025, A008454, A062503, A067259.
Similar sequences: A060476, A074661, A096432, A336064, A368715.

Select[Range[210], # == 1 || EvenQ[Max[FactorInteger[#][[;;, 2]]]] &]

lista(kmax) = for(k = 1, kmax, if(k == 1 || !(vecmax(factor(k)[,2])%2), print1(k, ", ")));

A060476 Let n = 2^e_2 * 3^e_3 * 5^e_5 * ... be the prime factorization of n; sequence gives n such that 1 + max{e_2, e_3, ...} is nonprime.

Original entry on oeis.org

1, 8, 24, 27, 32, 40, 54, 56, 72, 88, 96, 104, 108, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 200, 216, 224, 232, 243, 248, 250, 256, 264, 270, 280, 288, 296, 297, 312, 328, 343, 344, 351, 352, 360, 375, 376, 378, 384, 392, 408, 416, 424, 440, 456, 459, 472, 480

Offset: 1

N. J. A. Sloane, Sep 18 2008

nonn

The old entry with this sequence number was a duplicate of A005171.

The asymptotic density of this sequence is Sum_{c composite} (1/zeta(c) - 1/zeta(c-1)) = 0.1182437806... - Amiram Eldar, Oct 18 2020

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A096432, A074661.

Cf. A010051, A051903, A055229.

a060476 n = a060476_list !! (n-1)
a060476_list = filter ((== 0) . a010051' . (+ 1) . a051903) [1..]
-- Reinhard Zumkeller, Nov 30 2015

Join[{1}, Select[Range[500], !PrimeQ[1+Max[FactorInteger[#][[All, 2]]]]&]] (* Jean-François Alcover, Aug 02 2018 *)

isA060476(n) = if(n<2,1,!isprime(vecmax(factor(n)[,2])+1))

From Reinhard Zumkeller, Nov 30 2015: (Start)
A010051(A051903(a(n)+1)) = 1.
a(A055229(n)) > 1 for n > 1. (End)

A096432 Let n = 2^e_2 * 3^e_3 * 5^e_5 * ... be the prime factorization of n; sequence gives n such that 1 + max{e_2, e_3, ...} is a prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83

Offset: 1

N. J. A. Sloane, Sep 18 2008

nonn

The old entry with this sequence number was a duplicate of A004555.
Sequence is of positive density. - Charles R Greathouse IV, Dec 07 2012
The asymptotic density of this sequence is Sum_{p prime} (1/zeta(p) - 1/zeta(p-1)) = 0.8817562193... - Amiram Eldar, Oct 18 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..881 from R. J. Mathar)

Cf. A060476, A074661.

(Maple code for this entry and A074661)
M:=2000; ans1:=[]; ans2:=[];
for n from 1 to M do
t1:=op(2..-1, ifactors(n)); t2:=nops(t1);
m1:=0; for i from 1 to t2 do m1:=max(m1,t1[i][2]); od:
if isprime(1+m1) then ans1:=[op(ans1),n]; fi;
if isprime(m1) then ans2:=[op(ans2),n]; fi;
od:

Select[Range[2, 100], PrimeQ[1 + Max[FactorInteger[#][[;; , 2]]]] &] (* Amiram Eldar, Oct 18 2020 *)

isA096432(n) = if(n<2,0,isprime(vecmax(factor(n)[,2])+1))

A368715 Numbers that are not coprime to the maximal exponent in their prime factorization.

Original entry on oeis.org

4, 12, 16, 18, 20, 24, 27, 28, 36, 44, 48, 50, 52, 54, 60, 64, 68, 72, 76, 80, 84, 90, 92, 98, 100, 108, 112, 116, 120, 124, 126, 132, 135, 140, 144, 148, 150, 156, 160, 162, 164, 168, 172, 176, 180, 188, 189, 192, 196, 198, 204, 208, 212, 216, 220, 228, 234, 236, 240, 242, 244

Offset: 1

Amiram Eldar, Jan 04 2024

Subsequence of A137257 and first differs from it at n = 51.
Numbers k such that gcd(k, A051903(k)) > 1.
Includes all the nonsquarefree terms of A336064.
The asymptotic density of this sequence is 1 - 1/zeta(2) - Sum_{k>=2} (1/(f(k+1, k) * zeta(k+1)) - 1/(f(k, k) * zeta(k))) = 0.24998449199080279703..., where f(e, m) = Product_{primes p|m} ((1-1/p^e)/(1-1/p)).

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

Cf. A051903.
Subsequence of A013929 and A137257.
Similar sequences: A060476, A074661, A096432, A336064, A368714.

Select[Range[210], !CoprimeQ[#, Max[FactorInteger[#][[;;, 2]]]] &]

lista(kmax) = for(k = 2, kmax, if(gcd(k, vecmax(factor(k)[,2])) > 1, print1(k, ", ")));

A374588 Numbers whose maximum exponent in their prime factorization is a composite number.

Original entry on oeis.org

16, 48, 64, 80, 81, 112, 144, 162, 176, 192, 208, 240, 256, 272, 304, 320, 324, 336, 368, 400, 405, 432, 448, 464, 496, 512, 528, 560, 567, 576, 592, 624, 625, 648, 656, 688, 704, 720, 729, 752, 768, 784, 810, 816, 832, 848, 880, 891, 912, 944, 960, 976, 1008

Offset: 1

Amiram Eldar, Jul 12 2024

Subsequence of A322448 and first differs from it at n = 138: A322448(138) = 2592 = 2^5 * 3^4 is not a term of this sequence.

The asymptotic density of this sequence is d = Sum_{k composite} (1/zeta(k+1) - 1/zeta(k)) = 0.05296279266796920306... . The asymptotic density of this sequence within the nonsquarefree numbers (A013929) is d / (1 - 1/zeta(2)) = 0.13508404411123191108... .

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

Cf. A002808, A013661, A051903, A229099.
Complement of A074661 within A013929.
Subsequence of A322448 and A322449 \ {1}.
Similar sequences: A368714, A369937, A369938, A369939, A374589, A374590.

filter:= proc(n) local m;
  m:= max(ifactors(n)[2][..,2]);
  m > 1 and not isprime(m)
end proc:
select(filter, [$1..10000]); # Robert Israel, Jul 14 2024

Select[Range[1200], CompositeQ[Max[FactorInteger[#][[;; , 2]]]] &]

iscomposite(n) = n > 1 && !isprime(n);
is(n) = n > 1 && iscomposite(vecmax(factor(n)[, 2]));

cp's OEIS Frontend

A056594 Period 4: repeat [1,0,-1,0]; expansion of 1/(1 + x^2).

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A368714 Numbers whose maximal exponent in their prime factorization is even.

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A060476 Let n = 2^e_2 * 3^e_3 * 5^e_5 * ... be the prime factorization of n; sequence gives n such that 1 + max{e_2, e_3, ...} is nonprime.

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A096432 Let n = 2^e_2 * 3^e_3 * 5^e_5 * ... be the prime factorization of n; sequence gives n such that 1 + max{e_2, e_3, ...} is a prime.

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A368715 Numbers that are not coprime to the maximal exponent in their prime factorization.

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A374588 Numbers whose maximum exponent in their prime factorization is a composite number.

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