A016767 -id:A016767 - cp's OEIS frontend

A010693 Periodic sequence: Repeat 2,3.

2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3

Offset: 0

N. J. A. Sloane

a(n) = smallest prime divisor of n!! for n >= 2. For biggest prime divisor of n!! see A139421. - Artur Jasinski, Apr 21 2008
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-3, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=-charpoly(A,-2). - Milan Janjic, Jan 27 2010
Simple continued fraction of 1+sqrt(5/3) = A176020. - R. J. Mathar, Mar 08 2012
p(n) = a(n-1) is the Abelian complexity function of the Thue-Morse word A010060. - Nathan Fox, Mar 12 2013

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 466
G. Richomme, K. Saari, and L. Q. Zamboni, Abelian complexity in minimal subshifts, J. London Math. Soc. 83(1) (2011) 79-95.
G. Richomme, K. Saari, and L. Q. Zamboni, Abelian complexity in minimal subshifts, arXiv:0911.2914 [math.CO], 2009.
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A139421.

Cf. A026549 (partial products).

a010693 = (+ 2) . (`mod` 2)  -- Reinhard Zumkeller, Nov 27 2012
a010693_list = cycle [2,3]  -- Reinhard Zumkeller, Mar 29 2012

[2 + (n mod 2) : n in [0..100]]; // Wesley Ivan Hurt, Jul 24 2014

A010693:=n->2+(n mod 2): seq(A010693(n), n=0..100); # Wesley Ivan Hurt, Jul 24 2014

Table[5/2 - (-1)^n/2, {n, 0, 100}] or a = {}; Do[b = First[First[FactorInteger[n!! ]]]; AppendTo[a, b], {n, 2, 1000}]; a (* Artur Jasinski, Apr 21 2008 *)
2 + Mod[Range[0, 100], 2] (* Wesley Ivan Hurt, Jul 24 2014 *)
PadRight[{},120,{2,3}] (* Harvey P. Dale, Jan 20 2023 *)

a(n)=3 - (n+1)%2 \\ Charles R Greathouse IV, May 09 2016

a(n) = 5/2 - ((-1)^n)/2.
a(n) = 2 + (n mod 2) = A007395(n) + A000035(n). - Reinhard Zumkeller, Mar 23 2005
a(n) = A020639(A016767(n)) for n>0. - Reinhard Zumkeller, Jan 29 2009
From Jaume Oliver Lafont, Mar 20 2009: (Start)
G.f.: (2+3*x)/(1-x^2).
Linear recurrence: a(0)=2, a(1)=3, a(n)=a(n-2) for n>=2. (End)
a(n) = A001615(2n)/A001615(n) for n > 0. - Enrique Pérez Herrero, Jun 06 2012
a(n) = floor((n+1)*5/2) - floor((n)*5/2). - Hailey R. Olafson, Jul 23 2014
a(n) = 3 - ((n+1) mod 2). - Wesley Ivan Hurt, Jul 24 2014
E.g.f.: 2*cosh(x) + 3*sinh(x). - Stefano Spezia, Aug 04 2025

Definition rewritten by Bruno Berselli, Sep 30 2011

A155955 Triangle read by rows: T(n,k) = (k*n)^k, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 16, 1, 3, 36, 729, 1, 4, 64, 1728, 65536, 1, 5, 100, 3375, 160000, 9765625, 1, 6, 144, 5832, 331776, 24300000, 2176782336, 1, 7, 196, 9261, 614656, 52521875, 5489031744, 678223072849, 1, 8, 256, 13824, 1048576, 102400000, 12230590464

Offset: 0

Reinhard Zumkeller, Jan 31 2009

T(n,0)  = 1;
T(n,1)  = n for n > 0;
T(n,2)  = A016742(n) for n > 1;
T(n,3)  = A016767(n) for n > 2;
T(n,4)  = A016804(n) for n > 3;
T(n,5)  = A016853(n) for n > 4;
T(n,6)  = A016914(n) for n > 5;
T(n,7)  = A016987(n) for n > 6;
T(n,8)  = A017072(n) for n > 7;
T(n,9)  = A017169(n) for n > 8;
T(n,10) = A017278(n) for n > 9;
T(n,11) = A017399(n) for n > 10;
T(n,12) = A017532(n) for n > 11;
T(n,n)  = A062206(n).

			Triangle begins:
  1;
  1, 1;
  1, 2,  16;
  1, 3,  36,  729;
  1, 4,  64, 1728,  65536;
  1, 5, 100, 3375, 160000,  9765625;
  1, 6, 144, 5832, 331776, 24300000, 2176782336;
  ...

G. C. Greubel, Rows n=0..100 of triangle, flattened

Cf. A000312.

[[(n*k)^k: k in [0..n]]: n in [0..10]]; // G. C. Greubel, Sep 15 2018

Table[If[n == 0, 1, If[ k == 0, 1, (k*n)^k]], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Sep 15 2018 *)

for(n=0,10, for(k=0,n, print1((k*n)^k, ", "))) \\ G. C. Greubel, Sep 15 2018

A244725 a(n) = 5*n^3.

Original entry on oeis.org

0, 5, 40, 135, 320, 625, 1080, 1715, 2560, 3645, 5000, 6655, 8640, 10985, 13720, 16875, 20480, 24565, 29160, 34295, 40000, 46305, 53240, 60835, 69120, 78125, 87880, 98415, 109760, 121945, 135000, 148955, 163840, 179685, 196520, 214375, 233280, 253265

Offset: 0

Vincenzo Librandi, Jul 05 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. similar sequences of the type k*n^3: A000578 (k=1), A033431 (k=2), A117642 (k=3), A033430 (k=4), this sequence (k=5), A244726 (k=6), A244727 (k=7), A016743 (k=8), A244728 (k=9), A244729 (k=10), A016767 (k=27), A016803 (k=64), A016851 (k=125), A016911 (k=216), A016983 (k=343), A017067 (k=512), A017163 (k=729), A017271 (k=1000), A017391 (k=1331), A017523 (k=1728).

Magma
```
[5*n^3: n in [0..40]];
```

I:=[0,5,40,135]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]];

Table[5 n^3, {n, 0, 40}] (* or *) CoefficientList[Series[5 x (1 + 4 x + x^2)/(1 - x)^4, {x, 0, 40}], x]

a(n)=5*n^3 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: 5*x*(1 + 4*x + x^2)/(1 - x)^4.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3.

A097321 a(n) = (3n-1) 3n (3*n+1).

Original entry on oeis.org

24, 210, 720, 1716, 3360, 5814, 9240, 13800, 19656, 26970, 35904, 46620, 59280, 74046, 91080, 110544, 132600, 157410, 185136, 215940, 249984, 287430, 328440, 373176, 421800, 474474, 531360, 592620, 658416, 728910, 804264, 884640, 970200, 1061106, 1157520

Offset: 1

Ralf Stephan, Aug 07 2004

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

Cf. A069072, A069140, A002391, A016767.

[27*n^3-3*n: n in [1..40]]; // Vincenzo Librandi, Sep 07 2011

Table[27n^3-3n,{n,40}]  (* Harvey P. Dale, Mar 30 2011 *)

G.f.: 6x * (4x^2 + 19x + 4) / (1-x)^4.
Sum_{n>=1} 1/a(n) = (log(3) - 1)/2. - Amiram Eldar, Jul 04 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/2 - 2*log(2)/3. - Amiram Eldar, May 15 2022
E.g.f.: 3*exp(x)*x*(8 + 27*x + 9*x^2). - Stefano Spezia, Feb 20 2025

A243147 Least number k such that n^k + k^n is prime or 0 if no such k exists.

Original entry on oeis.org

1, 1, 2, 1, 24, 1, 54, 69, 2, 1, 3100, 1

Offset: 1

Derek Orr, May 30 2014

More terms given in links.
a(n) = 1 if and only if n + 1 is prime. Thus there are infinitely many nonzero entries.
For n in A016767, a(n) = 0 since n^k + k^n is factorable and will never be prime. Thus there are infinitely many zero entries.
If a(i) = j then a(j) <= i for all i and j not equal to 0.
a(n) and n must have opposite parity. If n is odd/even, a(n) must be even/odd, respectively.
Further, gcd(n, a(n)) = 1 for all n.

			3^1 + 1^3 = 4 is not prime. 3^2 + 2^3 = 17 is prime. So a(3) = 2.

H. Lifchitz and R. Lifchitz, PRP Top Records. Search for x^y+y^x
Derek Orr, Table for n, a(n) for n = 1..100 (unknown k-values are marked "unknown")

Cf. A016767.

a(n)=if(ispower(n)&&ispower(n)%3==0&&n%3==0,return(0));k=1;while(!ispseudoprime(n^k+k^n),k++);return(k)
vector(12, n, a(n))

A181968 a(n) = 54n^3 - 1.

Original entry on oeis.org

53, 431, 1457, 3455, 6749, 11663, 18521, 27647, 39365, 53999, 71873, 93311, 118637, 148175, 182249, 221183, 265301, 314927, 370385, 431999, 500093, 574991, 657017, 746495, 843749, 949103, 1062881, 1185407, 1317005, 1457999, 1608713, 1769471, 1940597, 2122415

Offset: 1

Arkadiusz Wesolowski, Apr 06 2012

a(n) is coprime to 27*n^3*(27*n^3 - 1) - 2 = A016767(n)*(A016767(n)-1) - 2.
x^3 + y^3 + z^3 = w^3 has infinitely many solutions, where every pair of elements x, y and z are coprime.
This follows from the identity a(n)^3 + (A016767(n)+1)^3 + (A016768(n)-A008588(n))^3 = (A016768(n)+A008585(n))^3 for n >= 1.

Wacław Sierpiński, Czym sie zajmuje teoria liczb. Warsaw: PW "Wiedza Powszechna", 1957, pp. 59-60.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A008585, A016767.

Magma
```
[ 54*n^3-1 : n in [1..34]];
```
Maple
```
seq(54*n^3-1, n=1..34);
```
Mathematica
```
Table[54*n^3 - 1, {n, 34}]
```
PARI
```
vector(34, n, 54*n^3-1)
```

For n >= 1, a(n) = 54*A000578(n) - 1 = 2*A016767(n) - 1.

G.f.: (-1 + 57*x + 213*x^2 + 55*x^3)/(1 - x)^4.

A229213 Square array of denominators of t(n,k) = (1+1/(k*n))^n, read by descending antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 16, 27, 4, 36, 216, 256, 5, 64, 729, 4096, 3125, 6, 100, 1728, 20736, 100000, 46656, 7, 144, 3375, 65536, 759375, 2985984, 823543, 8, 196, 5832, 160000, 3200000, 34012224, 105413504, 16777216, 9, 256

Offset: 1

Jean-François Alcover, Sep 16 2013

Limit(t(n,k), n -> infinity) = exp(1/k).
1st row = A000027
2nd row = A016742
3rd row = A016767
4th row = A016804
5th row = A016853
1st column = A000312
2nd column = A062971
3rd column = A091482
4th column = A091483

			Table of fractions begins:
   2,       3/2,        4/3,         5/4, ...
  9/4,     25/16,      49/36,       81/64, ...
64/27,   343/216,   1000/729,    2197/1728, ...
625/256, 6561/4096, 28561/20736, 83521/65536, ...
...
Table of denominators begins:
1,      2,     3,     4, ...
4,     16,    36,    64, ...
27,   216,   729,  1728, ...
256, 4096, 20736, 65536, ...
...
Triangle of antidiagonals begins:
1;
2, 4;
3, 16, 27;
4, 36, 216, 256;
...

Cf. A229212(numerators), A016742, A016767, A016804, A016853, A000312, A062971, A091482, A091483.

t[n_, k_] := (1+1/(k*n))^n; Table[t[n-k+1, k], {n, 1, 9}, {k, n, 1, -1}] // Flatten // Denominator

cp's OEIS Frontend

A010693 Periodic sequence: Repeat 2,3.

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A155955 Triangle read by rows: T(n,k) = (k*n)^k, 0 <= k <= n.

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A244725 a(n) = 5*n^3.

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A097321 a(n) = (3*n-1) * 3*n * (3*n+1).

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A243147 Least number k such that n^k + k^n is prime or 0 if no such k exists.

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PARI

A181968 a(n) = 54n^3 - 1.

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Magma

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A229213 Square array of denominators of t(n,k) = (1+1/(k*n))^n, read by descending antidiagonals.

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Comments

Examples

A097321 a(n) = (3n-1) 3n (3*n+1).