A017113 -id:A017113 - cp's OEIS frontend

A047608 Numbers that are congruent to {4, 5} mod 8.

4, 5, 12, 13, 20, 21, 28, 29, 36, 37, 44, 45, 52, 53, 60, 61, 68, 69, 76, 77, 84, 85, 92, 93, 100, 101, 108, 109, 116, 117, 124, 125, 132, 133, 140, 141, 148, 149, 156, 157, 164, 165, 172, 173, 180, 181, 188, 189, 196, 197, 204, 205, 212, 213, 220, 221, 228, 229

Offset: 1

N. J. A. Sloane

David Lovler, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Union of A017113 and A004770.

Select[Range[230], MemberQ[{4, 5}, Mod[#, 8]] &] (* Amiram Eldar, Dec 19 2021 *)

a(n) = 4*n - 3*(1 + (-1)^n)/2 \\ David Lovler, Aug 20 2022

G.f.: x*(4+x+3*x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Sep 22 2016
a(n) = 4n - 3*(1 + (-1)^n)/2 or a(n) = 4n - 3*((n-1) mod 2). - Heinz Ebert, Jul 12 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)-1)*Pi/16 - log(2)/4 + sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 19 2021
E.g.f.: 3 + ((8*x - 3)*exp(x) - 3*exp(-x))/2. - David Lovler, Aug 20 2022

A047610 Positive integers that are congruent to {1, 4, 5} mod 8.

Original entry on oeis.org

1, 4, 5, 9, 12, 13, 17, 20, 21, 25, 28, 29, 33, 36, 37, 41, 44, 45, 49, 52, 53, 57, 60, 61, 65, 68, 69, 73, 76, 77, 81, 84, 85, 89, 92, 93, 97, 100, 101, 105, 108, 109, 113, 116, 117, 121, 124, 125, 129, 132, 133, 137, 140, 141, 145, 148, 149, 153, 156, 157

Offset: 1

N. J. A. Sloane

Robert Israel, Table of n, a(n) for n = 1..9999
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A016813, A017113.

[n : n in [0..160] | n mod 8 in [1,4,5]]; // Vincenzo Librandi, May 07 2015

seq(op([8*i+1,8*i+4,8*i+5]),i=0..100); # Robert Israel, May 04 2015

a[1] := 1; a[2] := 4; a[3] := 5; a[n_] := a[n - 3] + 8; Table[a[n], {n, 10}] (* L. Edson Jeffery, May 04 2015 *)
Select[Range[0, 200], MemberQ[{1, 4, 5}, Mod[#, 8]] &] (* Vincenzo Librandi, May 07 2015 *)
LinearRecurrence[{1,0,1,-1},{1,4,5,9},60] (* Harvey P. Dale, Nov 21 2015 *)

is(n)=n%8==1||n%8>>1==2 \\ Charles R Greathouse IV, May 04 2015

a(n) = a(n-3) + 8, n > 3, with initial conditions a(1) = 1, a(2) = 4, a(3) = 5. - L. Edson Jeffery, May 04 2015
G.f.: x*(1+3*x)*(1+x^2)/(1-x-x^3+x^4). - Robert Israel, May 04 2015
A047610 = A016813 union A017113\{0}. - L. Edson Jeffery, May 06 2015

A062876 Numbers of lattice points corresponding to incrementally largest circle radii in A062875.

Original entry on oeis.org

4, 12, 20, 28, 44, 52, 68, 76, 92, 116, 124, 148, 164, 172, 188, 212, 236, 244, 268, 284, 292, 316, 332, 356, 388, 404, 412, 428, 436, 452, 508, 524, 548, 556, 596, 604, 628, 652, 668, 692, 716, 724, 764, 772, 788, 796, 844, 892, 908, 916, 932, 956, 964

Offset: 1

Eric W. Weisstein

For n = 1 and n >= 3, a(n) is the smallest nonsquarefree number divisible by prime(n). - David James Sycamore, Jun 15 2024

Ray Chandler, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Circle Lattice Points

Cf. A046112, A062875.
Cf. A017113, A111333.
Cf. A000040, A103929

[4] cat [4*NthPrime(n): n in [2..60]]; // Vincenzo Librandi, May 08 2015

Join[{4}, Table[4 Prime[n], {n, 2, 50}]] (* Vincenzo Librandi, May 08 2015 *)

a(n)=if(n>1,4*prime(n),4) \\ Charles R Greathouse IV, May 08 2015

from sympy import prime
def A062876(n): return prime(n)<<2 if n>1 else 4 # Chai Wah Wu, Aug 02 2024

a(n) = A017113(A111333(n)-1) = 8*A111333(n) - 4.

For n >= 2 a(n) = 4*A000040(n) (a term in A013929). - David James Sycamore, Jun 15 2024

Edited and extended by Ray Chandler, Jan 05 2012

A168398 a(n) = 4 + 8*floor((n-1)/2).

Original entry on oeis.org

4, 4, 12, 12, 20, 20, 28, 28, 36, 36, 44, 44, 52, 52, 60, 60, 68, 68, 76, 76, 84, 84, 92, 92, 100, 100, 108, 108, 116, 116, 124, 124, 132, 132, 140, 140, 148, 148, 156, 156, 164, 164, 172, 172, 180, 180, 188, 188, 196, 196, 204, 204, 212, 212, 220, 220, 228, 228

Offset: 1

Vincenzo Librandi, Nov 25 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A017113, A109613.

[4+8*Floor((n-1)/2): n in [1..70]]; // Vincenzo Librandi, Sep 18 2013

Table[4 + 8*Floor[(n - 1)/2], {n, 60}] (* Vincenzo Librandi, Sep 18 2013 *)
LinearRecurrence[{1,1,-1},{4,4,12},60] (* or *) With[{r=NestList[ #+8&,4,30]},Riffle[r,r]] (* Harvey P. Dale, Oct 18 2021 *)

a(n) = 8*n - a(n-1) - 8, with n>1, a(1)=4.
G.f.: 4*x*(1 + x^2)/((1+x)*(x-1)^2). - Vincenzo Librandi, Sep 18 2013
a(n) = a(n-1) +a(n-2) -a(n-3). - Vincenzo Librandi, Sep 18 2013
a(n) = 4 * A109613(n-1). - Bruno Berselli, Sep 18 2013
E.g.f.: 4*(1 + (x - 1)*cosh(x) + x*sinh(x)). - G. C. Greubel, Jul 20 2016

New definition by Vincenzo Librandi, Sep 18 2013

A346376 a(n) = n^4 + 14n^3 + 63n^2 + 98*n + 28.

Original entry on oeis.org

28, 204, 604, 1348, 2580, 4468, 7204, 11004, 16108, 22780, 31308, 42004, 55204, 71268, 90580, 113548, 140604, 172204, 208828, 250980, 299188, 354004, 416004, 485788, 563980, 651228, 748204, 855604, 974148, 1104580, 1247668, 1404204, 1575004, 1760908, 1962780

Offset: 0

Lamine Ngom, Jul 14 2021

The product of eight consecutive positive integers can always be expressed as the difference of two squares: x^2 - y^2.
This sequence gives the x-values for each product. The y-values are A017113(n+4).
a(n) is always divisible by 4. In addition, we have (a(n)+16)/4 belongs to A028387.
Are 4 and 8 the unique values of k such that the product of k consecutive integers is always distant to upper square by a square?

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A239035, A017113, A028387, A046691.

a(n) = A239035(n)^2 - A017113(n+4)^2.
a(n) = 4*(A028387(A046691(n+2)) - 4).
G.f.: 4*(7 + 16*x - 34*x^2 + 22*x^3 - 5*x^4)/(1 - x)^5. - Stefano Spezia, Jul 14 2021

A370596 Numbers k such that A007814(k) is a prime number.

Original entry on oeis.org

4, 8, 12, 20, 24, 28, 32, 36, 40, 44, 52, 56, 60, 68, 72, 76, 84, 88, 92, 96, 100, 104, 108, 116, 120, 124, 128, 132, 136, 140, 148, 152, 156, 160, 164, 168, 172, 180, 184, 188, 196, 200, 204, 212, 216, 220, 224, 228, 232, 236, 244, 248, 252, 260, 264, 268, 276

Offset: 1

Amiram Eldar, Feb 23 2024

Numbers whose binary representation has a prime number of trailing 0's.
a(n)-1 is the sequence of numbers whose binary representation has a prime number of trailing 1's.
Numbers of the form (2^(p+1))*k + 2^p = 2^p * (2*k + 1), where p is prime and k >= 0.
All the terms are divisible by 4.
The asymptotic density of this sequence is Sum_{p prime} 1/2^(p+1) = 0.20734125492555583012... = A051006 / 2.

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

Cf. A007814, A051006, A359794.

Subsequences: A017113, A051062.

Select[Range[300], PrimeQ[IntegerExponent[#, 2]] &]

PARI
```
is(n) = isprime(valuation(n, 2));
```

A017115 a(n) = (8*n + 4)^3.

Original entry on oeis.org

64, 1728, 8000, 21952, 46656, 85184, 140608, 216000, 314432, 438976, 592704, 778688, 1000000, 1259712, 1560896, 1906624, 2299968, 2744000, 3241792, 3796416, 4410944, 5088448, 5832000, 6644672, 7529536, 8489664, 9528128, 10648000, 11852352, 13144256, 14526784, 16003008

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002117, A016755, A016827, A017113.

[(8*n+4)^3: n in [0..35] ]; // Vincenzo Librandi, Jul 21 2011

LinearRecurrence[{4, -6, 4, -1},{64, 1728, 8000, 21952},24] (* Ray Chandler, Aug 04 2015 *)

G.f.: 64*(1+x)*(x^2 + 22*x + 1)/(x-1)^4. - R. J. Mathar, Jul 14 2016
From Amiram Eldar, Apr 25 2023: (Start)
a(n) = A017113(n)^3.
a(n) = 2^3*A016827(n) = 2^6*A016755(n).
Sum_{n>=0} 1/a(n) = 7*zeta(3)/512.
Sum_{n>=0} (-1)^n/a(n) = Pi^3/2048. (End)
E.g.f.: 64*exp(x)*(1 + 26*x + 36*x^2 + 8*x^3). - Stefano Spezia, May 27 2025

A017116 a(n) = (8*n + 4)^4.

Original entry on oeis.org

256, 20736, 160000, 614656, 1679616, 3748096, 7311616, 12960000, 21381376, 33362176, 49787136, 71639296, 100000000, 136048896, 181063936, 236421376, 303595776, 384160000, 479785216, 592240896, 723394816, 875213056, 1049760000, 1249198336, 1475789056, 1731891456

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A016756, A016828, A017113.

[(8*n+4)^4: n in [0..35] ]; // Vincenzo Librandi, Jul 21 2011

(8*Range[0,20]+4)^4 (* or *) LinearRecurrence[{5,-10,10,-5,1},{256,20736,160000,614656,1679616},20] (* Harvey P. Dale, Aug 05 2019 *)

G.f.: -256*(1 + 76*x + 230*x^2 + 76*x^3 + x^4)/(x-1)^5. - R. J. Mathar, May 08 2015
From Amiram Eldar, Apr 25 2023: (Start)
a(n) = A017113(n)^4.
a(n) = 2^4*A016828(n) = 2^8*A016756(n).
Sum_{n>=0} 1/a(n) = Pi^4/24576. (End)

A017117 a(n) = (8*n + 4)^5.

Original entry on oeis.org

1024, 248832, 3200000, 17210368, 60466176, 164916224, 380204032, 777600000, 1453933568, 2535525376, 4182119424, 6590815232, 10000000000, 14693280768, 21003416576, 29316250624, 40074642432, 53782400000, 71008211968, 92389579776, 118636749824, 150536645632, 188956800000

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A013663, A016757, A016829, A017113.

[(8*n+4)^5: n in [0..30] ]; // Vincenzo Librandi, Jul 21 2011

(8*Range[0,20]+4)^5 (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{1024,248832,3200000,17210368,60466176,164916224},20] (* Harvey P. Dale, Nov 24 2012 *)

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6); a(0)=1024, a(1)=248832, a(2)=3200000, a(3)=17210368, a(4)=60466176, a(5)=164916224. - Harvey P. Dale, Nov 24 2012
G.f.: 1024*(1+x)*(x^4 + 236*x^3 + 1446*x^2 + 236*x + 1) / (x-1)^6. - R. J. Mathar, May 08 2015
From Amiram Eldar, Apr 25 2023: (Start)
a(n) = A017113(n)^5.
a(n) = 2^5*A016829(n) = 2^10*A016757(n).
Sum_{n>=0} 1/a(n) = 31*zeta(5)/32768.
Sum_{n>=0} (-1)^n/a(n) = 5*Pi^5/1572864. (End)

A017118 a(n) = (8*n + 4)^6.

Original entry on oeis.org

4096, 2985984, 64000000, 481890304, 2176782336, 7256313856, 19770609664, 46656000000, 98867482624, 192699928576, 351298031616, 606355001344, 1000000000000, 1586874322944, 2436396322816, 3635215077376, 5289852801024, 7529536000000, 10509215371264, 14412774445056

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A016758, A016830, A017113.

[(8*n+4)^6: n in [0..20] ]; // Vincenzo Librandi, Jul 21 2011

(8*Range[0,30]+4)^6 (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{4096,2985984,64000000,481890304,2176782336,7256313856,19770609664},30] (* Harvey P. Dale, Jan 07 2016 *)

G.f.: -4096*(1 + 722*x + 10543*x^2 + 23548*x^3 + 10543*x^4 + 722*x^5 + x^6)/(x-1)^7. - R. J. Mathar, May 08 2015
From Amiram Eldar, Apr 25 2023: (Start)
a(n) = A017113(n)^6.
a(n) = 2^6*A016830(n) = 2^12*A016758(n).
Sum_{n>=0} 1/a(n) = Pi^6/3932160. (End)

cp's OEIS Frontend

A047608 Numbers that are congruent to {4, 5} mod 8.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A047610 Positive integers that are congruent to {1, 4, 5} mod 8.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A062876 Numbers of lattice points corresponding to incrementally largest circle radii in A062875.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Formula

Extensions

A168398 a(n) = 4 + 8*floor((n-1)/2).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A346376 a(n) = n^4 + 14*n^3 + 63*n^2 + 98*n + 28.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A370596 Numbers k such that A007814(k) is a prime number.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A017115 a(n) = (8*n + 4)^3.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A017116 a(n) = (8*n + 4)^4.

Views

Author

Keywords

A346376 a(n) = n^4 + 14n^3 + 63n^2 + 98*n + 28.