A201629 -id:A201629 - cp's OEIS frontend

A182140 Numbers n such that A060968(n) = A201629(n).

2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 255, 257

Offset: 1

José María Grau Ribas, Apr 14 2012

nonn

Includes prime numbers and the sequence A071700.

a(n) = A240960(n) for n <= 35. - Reinhard Zumkeller, Aug 05 2014

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

Cf. A060968, A201629, A071700, A240960.

a182140 n = a182140_list !! (n-1)
a182140_list = [x | x <- [1..], a060968 x == a201629 x]
-- Reinhard Zumkeller, Aug 05 2014

fa = FactorInteger; A060968[p_, s_] := Which[Mod[p, 4] == 1, p^( s - 1)*(p - 1), Mod[p, 4] == 3, p^(s - 1)*(p + 1), s == 1,2, True, 2^(s + 1)]; A060968[1] = 1; A060968[n_] := Product[A060968[fa[n][[i, 1]], fa[n][[i, 2]]], {i, Length[fa[n]]}]; A201629[n_] := Which[Mod[n, 4] == 1, (n - 1), Mod[n, 4] == 3, (n + 1), True, n]; Select[Range[1000], A060968[#] == A201629[#] &]

is(n)=my(f=factor(n)[, 1]); n*prod(i=if(n%2, 1, 2), #f, if(f[i]%4==1, 1-1/f[i], 1+1/f[i]))*if(n%4, 1, 2)==if(n%2,(n+1)\4*4,n) \\ Charles R Greathouse IV, Jul 03 2013

A212502 Composite numbers k that divide the imaginary part of (1+2i)^A201629(k).

Original entry on oeis.org

4, 8, 12, 16, 24, 32, 36, 48, 56, 64, 72, 96, 108, 112, 128, 132, 143, 144, 156, 168, 192, 216, 224, 256, 264, 272, 288, 312, 324, 336, 384, 392, 396, 399, 432, 448, 468, 496, 504, 512, 527, 528, 544, 552, 576, 624, 648, 672, 768, 779, 784, 792, 816, 864

Offset: 1

José María Grau Ribas, May 19 2012

nonn

If p is a prime number then p divides the imaginary part of (1+2i)^A201629(p).

The numbers of this sequence may be called Fermat pseudoprimes to base 1+2i.

Robert Israel, Table of n, a(n) for n = 1..10000
Jose María Grau, A. M. Oller-Marcen, Manuel Rodriguez and D. Sadornil, Fermat test with Gaussian base and Gaussian pseudoprimes, arXiv:1401.4708 [math.NT], 2014.

Cf. A201629, A213337, A212601.

A201629:= proc(n) if n::even then n elif n mod 4 = 1 then n-1 else n+1 fi end proc:
filter:= proc(n) not isprime(n) and type(Powmod(1+2*x, A201629(n), x^2+1, x) mod n, integer) end proc:
select(filter, [$2..1000]); # Robert Israel, Nov 06 2019

A201629[n_]:=Which[Mod[n,4]==3,n+1,Mod[n,4]==1,n-1,True,n]; Select[1+ Range[1000], ! PrimeQ[#] && Im[PowerMod[1 + 2I, A201629[#], #]] == 0 &]

Definition revised by José María Grau Ribas, Oct 12 2013

A213337 Odd composite numbers k that divide the imaginary part of (1+2i)^A201629(k).

Original entry on oeis.org

143, 399, 527, 779, 1501, 1679, 2407, 2627, 2703, 2737, 3239, 3289, 3599, 3827, 4031, 4033, 4879, 4991, 5183, 5291, 5719, 5921, 6479, 6601, 6721, 7739, 8321, 8687, 8903, 9361, 9503, 10153, 10439, 11537, 11663

Offset: 1

José María Grau Ribas, Jun 09 2012

nonn

The odd terms of A212502.

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

Cf. A201629, A212502.

t[n_]:=Which[Mod[n,4]==3,n+1,Mod[n,4]==1,n-1,True,n]; Select[1+ Range[1000], Mod[#,2]==1&&! PrimeQ[#] && Im[PowerMod[1 + 2I, t[#], #]] == 0 &]

A235864 G-Lehmer numbers: Composite numbers k such that A060968(k) divides A201629(k).

Original entry on oeis.org

15, 143, 255, 385, 3599, 5183, 11663, 32399, 34561, 36863, 51983, 57599, 65535, 97343, 121103, 147455, 176399, 186623, 195841, 359999, 435599, 685583, 1034881, 1040399, 1065023, 1192463, 1327103, 1742399, 2039183, 2108303, 2214143, 2585663, 2624399, 2782223, 3196943

Offset: 1

José María Grau Ribas, Jan 16 2014

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000
José María Grau, Antonio M. Oller-Marcén, Manuel Rodríguez, and Daniel Sadornil, Fermat test with Gaussian base and Gaussian pseudoprimes, Czechoslovak Mathematical Journal, Vol. 65 (2015), pp. 969-982; arXiv preprint, arXiv:1401.4708 [math.NT], 2014.

Cf. A060968, A201629, A235863, A182140.

fa=FactorInteger; phi[p_, s_] := Which[Mod[p, 4] == 1, p^(s-1)*(p-1), Mod[p, 4]==3, p^(s-1)*(p+1), s==1, 2, True, 2^(s+1)]; phi[1]=1; phi[n_] := Product[phi[fa[n][[i, 1]], fa[n][[i, 2]]], {i, Length[fa[n]]}]; Select[Range[1000], IntegerQ[FU[#]/phi[#]] && PrimeQ[#] == False &]

a(29)-a(35) from Amiram Eldar, Nov 24 2023

A235865 G-Carmichael numbers: Composite number such that A235863(n) divides A201629(n).

Original entry on oeis.org

4, 8, 12, 15, 16, 20, 24, 32, 36, 40, 48, 56, 60, 64, 72, 80, 96, 100, 105, 108, 112, 120, 128, 132, 143, 144, 156, 160, 168, 180, 192, 200, 216, 224, 240, 255, 256, 264, 272, 280, 288, 300, 312, 320, 324, 336, 360, 380, 384, 385, 392, 396, 399, 400, 432

Offset: 1

José María Grau Ribas, Jan 16 2014

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..1038
Jose María Grau, A. M. Oller-Marcen, Manuel Rodriguez and D. Sadornil, Fermat test with Gaussian base and Gaussian pseudoprimes, arXiv:1401.4708 [math.NT], 2014.

Cf. A235863, A201629.

FU[n_] := Which[Mod[n, 4] == 3, n + 1, Mod[n, 4] == 1, n - 1, True, n]; fa = FactorInteger; lam[1] = 1; lam[p_,s_] := Which[Mod[p, 4] == 3, p^(s - 1) (p + 1), Mod[p, 4] == 1, p^(s - 1) (p - 1), s ≥ 5, 2^(s -2), s > 1, 4, s == 1, 2]; lam[n_] := {aux = 1; Do[aux = LCM[aux, lam[fa[n][[i, 1]], fa[n][[i, 2]]]], {i, 1, Length[fa[n]]}]; aux}[[1]];Select[1+Range[1000], ! PrimeQ[#] && IntegerQ[FU[#]/lam[#]] &]

ok(n)={my(f=factor(n), r=n-kronecker( -4, n)); for(i=1, #f~, my([p, e]=f[i, ]); my(t=if(p==2, 2^max(e-2, min(e, 2)), p^(e-1)*if(p%4==1, p-1, p+1))); if(r%t, return(0)) ); n>1 && !isprime(n)} \\ Andrew Howroyd, Aug 06 2018

a(55) corrected by Andrew Howroyd, Aug 06 2018

A235863 Exponent of the multiplicative group G_n:={x+iy: x^2+y^2==1 (mod n); 0 <= x,y < n} where i=sqrt(-1).

Original entry on oeis.org

1, 2, 4, 4, 4, 4, 8, 4, 12, 4, 12, 4, 12, 8, 4, 4, 16, 12, 20, 4, 8, 12, 24, 4, 20, 12, 36, 8, 28, 4, 32, 8, 12, 16, 8, 12, 36, 20, 12, 4, 40, 8, 44, 12, 12, 24, 48, 4, 56, 20, 16, 12, 52, 36, 12, 8, 20, 28, 60, 4, 60, 32, 24, 16, 12, 12, 68, 16, 24, 8, 72

Offset: 1

José María Grau Ribas, Jan 16 2014

nonn

From Jianing Song, Nov 05 2019: (Start)
Exponent of the group G is the least e > 0 such that x^e = 1 for every x in G, where 1 is the identity element.
Also the exponent of O(2,Z_n) or SO(2,Z_n). O(2,Z_n) is the group of 2 X 2 matrices A over Z_n such that A*A^T = E = [1,0;0,1]; SO(2,Z_n) is the group of 2 X 2 matrices A over Z_n such that A*A^T = E = [1,0;0,1] and det(A) = 1. Note that G_n is isomorphic to SO(2,Z_n) by the mapping x+yi <-> [x,y;-y,x]. See A060698 for the group structure of SO(2,Z_n) and A182039 for the group structure of O(2,Z_n). (End)

Andrew Howroyd, Table of n, a(n) for n = 1..10000
José María Grau, A. M. Oller-Marcén, Manuel Rodriguez and Daniel Sadornil, Fermat test with Gaussian base and Gaussian pseudoprimes, arXiv:1401.4708 [math.NT], 2014.
José María Grau, A. M. Oller-Marcén, Manuel Rodriguez and Daniel Sadornil, Fermat test with Gaussian base and Gaussian pseudoprimes, Czechoslovak Mathematical Journal 65(140), (2015) pp. 969-982.

                                   (Z/nZ)*    ------    G_n
Order:                             A000010    ------  A060968.
Exponent:                          A002322    ------  this sequence.
                                     n-1      ------  A201629.
Carmichael/G-Carmichael numbers:   A002997    ------  A235865.
Lehmer /G-Lehmer numbers:          unknown    ------  A235864.
Cyclic/G-cyclic numbers:           A003277    ------  A235866.
n such that the group is cyclic:   A033948    ------  A235868.
Cf. A060698, A182039.

fa=FactorInteger; lam[1]=1;lam[p_, s_] := Which[Mod[p, 4] == 3, p ^ (s - 1 ) (p + 1) , Mod[p, 4] == 1, p ^ (s - 1 ) (p - 1)  , s ≥ 5, 2 ^ (s - 2 ), s > 1, 4, s == 1, 2];lam[n_] := {aux = 1; Do[aux = LCM[aux, lam[fa[n][[i, 1]], fa[n][[i, 2]]]], {i, 1, Length[fa[n]]}]; aux}[[1]] ; Array[lam, 100]

a(n)={my(f=factor(n)); lcm(vector(#f~, i, my([p,e]=f[i,]); if(p==2, 2^max(e-2, min(e,2)), p^(e-1)*if(p%4==1, p-1, p+1))))} \\ Andrew Howroyd, Aug 06 2018

a(2) = 2, a(4) = a(8) = a(16) = 4, a(2^e) = 2^(e-2) for e >= 5; a(p^e) = (p-1)*p^(e-1) if p == 1 (mod 4) and (p+1)*p^(e-1) if p == 1 (mod 4). - Jianing Song, Nov 05 2019

If gcd(n,m)=1 then a(nm) = lcm(a(n), a(m)).

A212831 a(4n) = 2n, a(2n+1) = 2n+1, a(4n+2) = 2n+2.

Original entry on oeis.org

0, 1, 2, 3, 2, 5, 4, 7, 4, 9, 6, 11, 6, 13, 8, 15, 8, 17, 10, 19, 10, 21, 12, 23, 12, 25, 14, 27, 14, 29, 16, 31, 16, 33, 18, 35, 18, 37, 20, 39, 20, 41, 22, 43, 22, 45, 24, 47, 24, 49, 26, 51, 26, 53, 28, 55, 28, 57, 30, 59, 30, 61, 32, 63, 32, 65, 34, 67, 34, 69, 36, 71, 36, 73, 38, 75

Offset: 0

Paul Curtz, Aug 14 2012

First differences: (1, 1, 1, -1, 3, -1, 3, -3, 5,...) = (1, A186422).
Second differences: (0, 0, -2, 4, -4, 4, -6, 8, ...)  = (-1)^(n+1) * A201629(n).
Interleave the terms with even indices of the companion A215495 and this one to get (A215495(0), A212831(0), A215495(2), A212831(2),...) = (1, 0, 1, 2, 3, 2, 3, 4, 5, 4,...) = A106249, up to the initial term = A083219 = A083220/2.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).

Cf. A214282, A129756.

[(1/4)*((1 +(-1)^n)*(1 - (-1)^Floor(n/2)) + (3 -(-1)^n)*n): n in [0..50]]; // G. C. Greubel, Apr 25 2018

a[n_] := (1/4)*((-(1 + (-1)^n))*(-1 + (-1)^Floor[n/2]) - (-3 + (-1)^n)*n ); Table[a[n], {n, 0, 84}] (* Jean-François Alcover, Sep 18 2012 *)
LinearRecurrence[{0,1,0,1,0,-1},{0,1,2,3,2,5},80] (* Harvey P. Dale, May 29 2016 *)

A212831(n)=if(bittest(n,0), n, n\2+bittest(n,1)) \\ M. F. Hasler, Oct 21 2012

for(n=0,50, print1((1/4)*((1 +(-1)^n)*(1 - (-1)^floor(n/2)) + (3 -(-1)^n)*n), ", ")) \\ G. C. Greubel, Apr 25 2018

a(n) + A215495(n) = A043547(n).
a(n) = -A214283(n)/A000108([n/2]).
a(n+1) = (A186421(n)=0,1,2,1,4,...) + 1.
a(2*n) = A052928(n+1).
a(n+2) - a(n) = 2, 2, 0, 2. (period 4).
a(n) = a(n-2) +a(n-4) -a(n-6); also holds for A215495(n).
G.f.: x*(1+2*x+2*x^2+x^4) / ( (x^2+1)*(x-1)^2*(1+x)^2 ). - R. J. Mathar, Aug 21 2012
a(n) = (1/4)*((1 +(-1)^n)*(1 - (-1)^floor(n/2)) + (3 -(-1)^n)*n). - G. C. Greubel, Apr 25 2018

Corrected and edited by M. F. Hasler, Oct 21 2012

A212601 Intersection of A001567 and A212502.

Original entry on oeis.org

4033, 6601, 8321, 15841, 25761, 29341, 41041, 46657, 75361, 115921, 162401, 172081, 252601, 294409, 314821, 332949, 401401, 410041, 488881, 530881, 552721, 642001, 721801, 873181, 934021, 1004653, 1207361, 1461241, 1876393, 1909001, 2081713, 2085301, 2113921

Offset: 1

José María Grau Ribas, May 22 2012

nonn

Only 1 (mod 4) numbers have been found.

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

Cf. A201629.
Cf. A001567 (pseudoprimes to base 2).
Cf. A212502 (pseudoprimes to base 1+2i).

t[n_] := Which[Mod[n, 4] == 3, n + 1, Mod[n, 4] == 1, n - 1,  True,  n]; Select[1 + Range[99000], PowerMod[2, # - 1, #] == 1 && !PrimeQ[#] && Im[PowerMod[1 + 2I, t[#], #]] == 0 &]

A182221 Composite numbers in A182140 but not in A071700.

Original entry on oeis.org

255, 385, 34561, 65535, 147455, 195841, 1034881, 4070401, 4746241, 5040001, 16675201, 22704001, 36067201, 47013121, 83623935, 136967041, 168720001, 271878145, 549141119, 613092481, 836567041, 1039779841, 1049759999, 1548072961, 2556902401, 2646067201

Offset: 1

José María Grau Ribas, Apr 19 2012

nonn

A182140 includes the prime numbers and A071700.

J. M. Grau, A. M. Oller-Marcen, M. Rodríguez, D. Sadornil, Fermat test with gaussian base and Gaussian pseudoprimes, arXiv preprint arXiv:1401.4708 [math.NT], 2014.

Cf. A182140, A071700, A060968, A201629.

fa=FactorInteger;isA071700[n_]:=Length@fa[n]==2&&fa[n][[1,2]]==fa[n][[2,2]]==1&&Mod[Sqrt[n+1],4]==0; A060968[p_, s_] := Which[Mod[p, 4] == 1, p^(s-1)*(p-1), Mod[p, 4]==3, p^(s-1)*(p+1), s==1, 2, True, 2^(s+1)]; A060968[1]=1; A060968[n_] := Product[A060968[fa[n][[i, 1]], fa[n][[i, 2]]], {i, Length[fa[n]]}]; A201629[n_]:=Which[Mod[n, 4]==1, (n-1), Mod[n, 4]==3, (n+1), True, n];
Select[Range[10000], !isA071700[#]&&!PrimeQ[#]&&A060968[#]==A201629[#]&]

isA071700(n)=my(k=sqrtint(n\16)); n==16*k^2+32*k+15 && isprime(4*k+3) && isprime(4*k+5)
is(n)=if(isprime(n),return(0)); my(f=factor(n)[, 1]); n*prod(i=if(n%2, 1, 2), #f, if(f[i]%4==1, 1-1/f[i], 1+1/f[i]))*if(n%4, 1, 2)==if(n%2,(n+1)\4*4,n) && !isA071700(n) \\ Charles R Greathouse IV, Jul 03 2013

a(15)-a(23) from Charles R Greathouse IV, Jul 03 2013

a(24)-a(26) from Charles R Greathouse IV, Jul 05 2013

A185342 Triangle of successive recurrences in columns of A117317(n).

Original entry on oeis.org

2, 4, -4, 6, -12, 8, 8, -24, 32, -16, 10, -40, 80, -80, 32, 12, -60, 160, -240, 192, -64, 14, -84, 280, -560, 672, -448, 128, 16, -112, 448, -1120, 1792, -1792, 1024, -256, 18, -144, 672, -2016, 4032, -5376, 4608, -2304, 512, 20, -180, 960, -3360, 8064

Offset: 0

Paul Curtz, Jan 26 2012

A117317 (A):
1
2    1
4    5   1
8   16   9   1
16  44  41  14   1
32 112 146  85  20  1
64 272 456 377 155 27 1
have for their columns successive signatures
(2) (4,-4) (6,-12,8) (8,-24, 32, -16) (10,-40,80,-80,32) i.e. a(n).
Take based on abs(A133156) (B):
1
2   0
4   1  0
8   4  0 0
16 12  1 0 0
32 32  6 0 0 0
64 80 24 1 0 0 0.
The recurrences of successive columns are also a(n). a(n) columns: A005843(n+1), A046092(n+1), A130809, A130810, A130811, A130812, A130813.
A053220 + A001787 = A014480.

			Triangle T(n,k),for 1<=k<=n, begins :
2                                         (1)
4    -4                                   (2)
6   -12   8                               (3)
8   -24  32   -16                         (4)
10  -40  80   -80   32                    (5)
12  -60 160  -240  192   -64              (6)
14  -84 280  -560  672  -448  128         (7)
16 -112 448 -1120 1792 -1792 1024 -256    (8)
Successive rows can be divided by A171977.

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. For (A): A053220, A056243. For (B): A000079, A001787, A001788, A001789. For A193862: A115068 (a Coxeter group). For (2): A014480 (also (3),(4),(5),..); also A053220 and A001787.

Cf. A007318.

Table[(-1)*Binomial[n, k]*(-2)^k, {n, 1, 20}, {k, 1, n}] // Flatten (* G. C. Greubel, Jun 27 2017 *)

for(n=1,20, for(k=1,n, print1((-2)^(k+1)*binomial(n,k)/2, ", "))) \\ G. C. Greubel, Jun 27 2017

Take A133156(n) without 1's or -1's double triangle (C)=
2
4
8      -4
16    -12
32    -32   6
64    -80  24
128  -192  80   -8
256  -448 240  -40
512 -1024 672 -160 10;
a(n) is increasing odd diagonals and increasing (sign changed) even diagonals. Rows sum of (C) = A201629 (?) Another link between Chebyshev polynomials and cos( ).
Absolute values: A013609(n) without 1's. Also 2*A193862 = (2*A002260)*A135278.
T(n,k) = T(n-1,k) - 2*T(n-1,k-1) for k>1, T(n,1) = 2*n = 2*T(n-1,1) - T(n-2,1). - Philippe Deléham, Feb 11 2012
T(n,k) = (-1)* Binomial(n,k)*(-2)^k, 1<=k<=n. - Philippe Deléham, Feb 11 2012

cp's OEIS Frontend

A182140 Numbers n such that A060968(n) = A201629(n).

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A212502 Composite numbers k that divide the imaginary part of (1+2i)^A201629(k).

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A213337 Odd composite numbers k that divide the imaginary part of (1+2i)^A201629(k).

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A235864 G-Lehmer numbers: Composite numbers k such that A060968(k) divides A201629(k).

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A235865 G-Carmichael numbers: Composite number such that A235863(n) divides A201629(n).

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A235863 Exponent of the multiplicative group G_n:={x+iy: x^2+y^2==1 (mod n); 0 <= x,y < n} where i=sqrt(-1).

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A212831 a(4*n) = 2*n, a(2*n+1) = 2*n+1, a(4*n+2) = 2*n+2.

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A212601 Intersection of A001567 and A212502.

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A212831 a(4n) = 2n, a(2n+1) = 2n+1, a(4n+2) = 2n+2.