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Search: a016767 -id:a016767
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A010693
Periodic sequence: Repeat 2,3.
+10
27
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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
LINKS
Table of n, a(n) for n=0..101.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 466
G. Richomme, K. Saari, L. Q. Zamboni, Abelian complexity in minimal subshifts, J. London Math. Soc. 83(1) (2011) 79-95.
G. Richomme, K. Saari, 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).
FORMULA
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
MAPLE
A010693:=n->2+(n mod 2): seq(A010693(n), n=0..100); # Wesley Ivan Hurt, Jul 24 2014
MATHEMATICA
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 *)
PROG
(Haskell)
a010693 = (+ 2) . (`mod` 2) -- Reinhard Zumkeller, Nov 27 2012
a010693_list = cycle [2, 3] -- Reinhard Zumkeller, Mar 29 2012
(Magma) [2 + (n mod 2) : n in [0..100]]; // Wesley Ivan Hurt, Jul 24 2014
(PARI) a(n)=3 - (n+1)%2 \\ Charles R Greathouse IV, May 09 2016
CROSSREFS
Cf. A139421.
Cf. A026549 (partial products).
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
EXTENSIONS
Definition rewritten by Bruno Berselli, Sep 30 2011
STATUS
approved
A155955
Triangle read by rows: T(n,k) = (k*n)^k, 0 <= k <= n.
+10
15
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
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
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).
LINKS
G. C. Greubel, Rows n=0..100 of triangle, flattened
EXAMPLE
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;
...
MATHEMATICA
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 *)
PROG
(PARI) for(n=0, 10, for(k=0, n, print1((k*n)^k, ", "))) \\ G. C. Greubel, Sep 15 2018
(Magma) [[(n*k)^k: k in [0..n]]: n in [0..10]]; // G. C. Greubel, Sep 15 2018
CROSSREFS
Cf. A000312.
KEYWORD
nonn,tabl
AUTHOR
Reinhard Zumkeller, Jan 31 2009
STATUS
approved
A244725
a(n) = 5*n^3.
+10
11
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
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.
MATHEMATICA
Table[5 n^3, {n, 0, 40}] (* or *) CoefficientList[Series[5 x (1 + 4 x + x^2)/(1 - x)^4, {x, 0, 40}], x]
PROG
(Magma) [5*n^3: n in [0..40]] /* or */ 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]];
(PARI) a(n)=5*n^3 \\ Charles R Greathouse IV, Oct 07 2015
CROSSREFS
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).
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Jul 05 2014
STATUS
approved
A097321
a(n) = (3*n-1) * 3*n * (3*n+1).
+10
6
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
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).
FORMULA
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
MATHEMATICA
Table[27n^3-3n, {n, 40}] (* Harvey P. Dale, Mar 30 2011 *)
PROG
(Magma) [27*n^3-3*n: n in [1..40]]; // Vincenzo Librandi, Sep 07 2011
CROSSREFS
Cf. A069072, A069140, A002391, A016767.
KEYWORD
nonn,easy
AUTHOR
Ralf Stephan, Aug 07 2004
STATUS
approved
A243147
Least number k such that n^k + k^n is prime or 0 if no such k exists.
+10
2
1, 1, 2, 1, 24, 1, 54, 69, 2, 1, 3100, 1
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
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.
LINKS
Table of n, a(n) for n=1..12.
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")
EXAMPLE
3^1 + 1^3 = 4 is not prime. 3^2 + 2^3 = 17 is prime. So a(3) = 2.
PROG
(PARI) 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))
CROSSREFS
Cf. A016767.
KEYWORD
nonn,hard,more
AUTHOR
Derek Orr, May 30 2014
STATUS
approved
A181968
54n^3 - 1.
+10
1
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
REFERENCES
Wacław Sierpiński, Czym sie zajmuje teoria liczb. Warsaw: PW "Wiedza Powszechna", 1957, pp. 59-60.
LINKS
Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
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.
MAPLE
seq(54*n^3-1, n=1..34);
MATHEMATICA
Table[54*n^3 - 1, {n, 34}]
PROG
(Magma) [ 54*n^3-1 : n in [1..34]]
(PARI) vector(34, n, 54*n^3-1)
CROSSREFS
Cf. A008585, A016767.
KEYWORD
easy,nonn
AUTHOR
Arkadiusz Wesolowski, Apr 06 2012
STATUS
approved
A229213
Square array of denominators of t(n,k) = (1+1/(k*n))^n, read by descending antidiagonals.
+10
1
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
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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
LINKS
Table of n, a(n) for n=1..38.
EXAMPLE
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;
...
MATHEMATICA
t[n_, k_] := (1+1/(k*n))^n; Table[t[n-k+1, k], {n, 1, 9}, {k, n, 1, -1}] // Flatten // Denominator
CROSSREFS
Cf. A229212(numerators), A016742, A016767, A016804, A016853, A000312, A062971, A091482, A091483.
KEYWORD
frac,tabl,nonn
AUTHOR
Jean-François Alcover, Sep 16 2013
STATUS
approved
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