A000351 -id:A000351 - cp's OEIS frontend

A223233 T(n,k)=Rolling icosahedron footprints: number of nXk 0..11 arrays starting with 0 where 0..11 label vertices of an icosahedron and every array movement to a horizontal, diagonal or antidiagonal neighbor moves along an icosahedral edge.

1, 5, 12, 25, 65, 144, 125, 785, 845, 1728, 625, 7445, 25225, 10985, 20736, 3125, 75665, 492365, 812225, 142805, 248832, 15625, 753005, 11043445, 32837285, 26157625, 1856465, 2985984, 78125, 7540985, 236027705, 1697263985, 2191464605, 842416625

Offset: 1

R. H. Hardin Mar 18 2013

Table starts
............1.............5................25.................125
...........12............65...............785................7445
..........144...........845.............25225..............492365
.........1728.........10985............812225............32837285
........20736........142805..........26157625..........2191464605
.......248832.......1856465.........842416625........146259564725
......2985984......24134045.......27130395625.......9761484584045
.....35831808.....313742585......873746350625.....651489782832965
....429981696....4078653605....28139386665625...43480983274973885
...5159780352...53022496865...906241361740625.2901957882023749205
..61917364224..689292459245.29185902861015625
.743008370688.8960801970185

			Some solutions for n=3 k=4
..0..6..0..5....0..5..6..5....0..7..0..1....0..1..3..1....0..1..0..7
..0..6.10..5....0..5..6..5....3..7..0..7....3..7..3..9....0..7..0..7
..0..5.10..4....6..2..6..5....3..7..5..7....3..9.11..7....3..1..3..7
Vertex neighbors:
0 -> 1 2 5 6 7
1 -> 0 2 3 7 8
2 -> 0 1 4 6 8
3 -> 1 7 8 9 11
4 -> 2 6 8 9 10
5 -> 0 6 7 10 11
6 -> 0 2 4 5 10
7 -> 0 1 3 5 11
8 -> 1 2 3 4 9
9 -> 3 4 8 10 11
10 -> 4 5 6 9 11
11 -> 3 5 7 9 10

R. H. Hardin, Table of n, a(n) for n = 1..97

Column 1 is A001021(n-1)
Column 2 is 5*13^(n-1)
Row 1 is A000351(n-1)

Empirical for column k:
k=1: a(n) = 12*a(n-1)
k=2: a(n) = 13*a(n-1)
k=3: a(n) = 35*a(n-1) -90*a(n-2)
k=4: a(n) = 73*a(n-1) -423*a(n-2) +351*a(n-3)
k=5: [order 11]
k=6: [order 26]
Empirical for row n:
n=1: a(n) = 5*a(n-1)
n=2: a(n) = 7*a(n-1) +30*a(n-2) for n>3
n=3: a(n) = 18*a(n-1) +103*a(n-2) -552*a(n-3) +540*a(n-4) for n>5
n=4: a(n) = [order 12] for n>13

A303934 Number of ways to write 2*n as p + 2^k + 5^m with p prime and 2^k + 5^m squarefree, where k and m are nonnegative integers.

Original entry on oeis.org

0, 1, 1, 3, 3, 2, 2, 3, 3, 4, 3, 5, 4, 4, 3, 4, 5, 7, 4, 7, 4, 8, 7, 6, 7, 6, 5, 5, 5, 7, 5, 8, 5, 5, 8, 6, 9, 9, 6, 8, 6, 6, 7, 8, 4, 7, 8, 7, 3, 10, 6, 7, 8, 7, 7, 9, 5, 8, 7, 6, 5, 5, 6, 3, 11, 7, 9, 12, 8, 12, 10, 11, 11, 9, 7, 9, 7, 8, 8, 11, 7, 11, 8, 9, 15, 11, 8, 9, 8, 9

Offset: 1

Zhi-Wei Sun, May 03 2018

nonn

Conjecture: a(n) > 0 for all n > 1.
This has been verified for all n = 2..10^10.
Note that a(n) <= A303821(n).

			a(2) = 1 since 2*2 = 2 + 2^0 + 5^0 with 2 prime and 2^0 + 5^0 squarefree.
a(3) = 1 since 2*3 = 3 + 2^1 + 5^0 with 3 prime and 2^1 + 5^0 squarefree.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Mixed sums of primes and other terms, in: Additive Number Theory (edited by D. Chudnovsky and G. Chudnovsky), pp. 341-353, Springer, New York, 2010.
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT], 2012-2017.)

Cf. A000040, A000079, A000351, A005117, A118955, A156695, A273812, A302982, A302984, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303543, A303601, A303637, A303639, A303656, A303660, A303702, A303821, A303932, A304034, A304081, A304122.

tab={};Do[r=0;Do[If[SquareFreeQ[2^k+5^m]&&PrimeQ[2n-2^k-5^m],r=r+1],{k,0,Log[2,2n-1]},{m,0,Log[5,2n-2^k]}];tab=Append[tab,r],{n,1,90}];Print[tab]

A339685 a(n) = Sum_{d|n} 5^(d-1).

Original entry on oeis.org

1, 6, 26, 131, 626, 3156, 15626, 78256, 390651, 1953756, 9765626, 48831406, 244140626, 1220718756, 6103516276, 30517656381, 152587890626, 762939846906, 3814697265626, 19073488282006, 95367431656276, 476837167968756, 2384185791015626, 11920929003987656

Offset: 1

Ilya Gutkovskiy, Dec 12 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Column 5 of A308813.
Cf. A000351, A295506.
Sums of the form Sum_{d|n} q^(d-1): A034729 (q=2), A034730 (q=3), A113999 (q=10), A339684 (q=4), this sequence (q=5), A339686 (q=6), A339687 (q=7), A339688 (q=8), A339689 (q=9).

A339685:= func< n | (&+[5^(d-1): d in Divisors(n)]) >;
[A339685(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024

Table[Sum[5^(d - 1), {d, Divisors[n]}], {n, 1, 24}]
nmax = 24; CoefficientList[Series[Sum[x^k/(1 - 5 x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, 5^(d-1)); \\ Michel Marcus, Dec 13 2020

def A339685(n): return sum(5^(k-1) for k in (1..n) if (k).divides(n))
[A339685(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024

G.f.: Sum_{k>=1} x^k / (1 - 5*x^k).
G.f.: Sum_{k>=1} 5^(k-1) * x^k / (1 - x^k).
a(n) ~ 5^(n-1). - Vaclav Kotesovec, Jun 05 2021

A075500 Stirling2 triangle with scaled diagonals (powers of 5).

Original entry on oeis.org

1, 5, 1, 25, 15, 1, 125, 175, 30, 1, 625, 1875, 625, 50, 1, 3125, 19375, 11250, 1625, 75, 1, 15625, 196875, 188125, 43750, 3500, 105, 1, 78125, 1984375, 3018750, 1063125, 131250, 6650, 140, 1, 390625, 19921875

Offset: 1

Wolfdieter Lang, Oct 02 2002

This is a lower triangular infinite matrix of the Jabotinsky type. See the Knuth reference given in A039692 for exponential convolution arrays.

The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(5*z) - 1)*x/5) - 1.

			[1]; [5,1]; [25,15,1]; ...; p(3,x) = x(25 + 15*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*     1
*     5       1
*    25      15       1
*   125     175      30       1
*   625    1875     625      50      1
*  3125   19375   11250    1625     75    1
* 15625  196875  188125   43750   3500  105   1
* 78125 1984375 3018750 1063125 131250 6650 140 1
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 1-7 are A000351, A016164, A075911-A075915. Row sums are A005011(n-1).

Cf. A075499, A075501.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> 5^n, 9); # Peter Luschny, Jan 28 2016

Flatten[Table[5^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 10;
M = BellMatrix[5^#&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

for(n=1, 11, for(m=1, n, print1(5^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

a(n, m) = (5^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*5)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 5m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-5k*x), m >= 1.
E.g.f. for m-th column: (((exp(5x)-1)/5)^m)/m!, m >= 1.

A152187 a(n) = 3a(n-1) + 5a(n-2), with a(0)=1, a(1)=5.

Original entry on oeis.org

1, 5, 20, 85, 355, 1490, 6245, 26185, 109780, 460265, 1929695, 8090410, 33919705, 142211165, 596232020, 2499751885, 10480415755, 43940006690, 184222098845, 772366329985, 3238209484180, 13576460102465, 56920427728295

Offset: 0

Philippe Deléham, Nov 28 2008

Unsigned version of A152185.
From Johannes W. Meijer, Aug 01 2010: (Start)
The a(n) represent the number of n-move routes of a fairy chess piece starting in a given side square (m = 2, 4, 6 and 8) on a 3 X 3 chessboard. This fairy chess piece behaves like a king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.
The sequence above corresponds to 24 red king vectors, i.e., A[5] vectors, with decimal values 27, 30, 51, 54, 57, 60, 90, 114, 120, 147, 150, 153, 156, 177, 180, 210, 216, 240, 282, 306, 312, 402, 408 and 432. These vectors lead for the corner squares to A015523 and for the central square to A179606.
This sequence belongs to a family of sequences with g.f. (1+2*x)/(1 - 3*x - k*x^2). Red king sequences that are members of this family are A007483 (k=2), A108981 (k=4), A152187 (k=5; this sequence), A154964 (k=6), A179602 (k=7) and A179598 (k=8). We observe that there is no red king sequence for k=3. Other members of this family are A036563 (k=-2), A054486 (k=-1), A084244 (k=0), A108300 (k=1) and A000351 (k=10).
Inverse binomial transform of A015449 (without the first leading 1).
(End)

Index entries for linear recurrences with constant coefficients, signature (3, 5).

Cf. A015523, A072263, A072264, A179606, A197189.

LinearRecurrence[{3,5},{1,5},40] (* Harvey P. Dale, May 03 2013 *)

G.f.: (1+2*x)/(1 - 3*x - 5*x^2).
Lim_{k->infinity} a(n+k)/a(k) = (A072263(n) + A015523(n)*sqrt(29))/2. - Johannes W. Meijer, Aug 01 2010
G.f.: G(0)*(1+2*x)/(2-3*x), where G(k) = 1 + 1/(1 - x*(29*k-9)/(x*(29*k+20) - 6/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 17 2013

A158960 Numerator of Hermite(n, 1/5).

Original entry on oeis.org

1, 2, -46, -292, 6316, 71032, -1436936, -24183472, 454560656, 10582510112, -183387274976, -5658029605952, 89546942024896, 3573911647620992, -51057689020940416, -2603853531376575232, 33085559702952161536, 2149253944507164508672

Offset: 0

N. J. A. Sloane, Nov 12 2009

The denominators are 5^n = A000351(n) (conjectured). - M. F. Hasler, Feb 16 2014

			Numerators of 1, 2/5, -46/25, -292/125, 6316/625, 71032/3125, -1436936/15625,..

Vincenzo Librandi, Table of n, a(n) for n = 0..200
DLMF Digital library of mathematical functions, Table 18.9.1 for H_n(x)

[Numerator((&+[(-1)^k*Factorial(n)*(2/5)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jun 02 2018

A158960 := proc(n)
    orthopoly[H](n,1/5) ;
    numer(%) ;
end proc: # R. J. Mathar, Feb 16 2014

Numerator[Table[HermiteH[n,1/5],{n,0,50}]] (* Vladimir Joseph Stephan Orlovsky, Mar 23 2011*)

A158960 = n->numerator(polhermite(n,1/5)) \\ M. F. Hasler, Feb 16 2014

D-finite with recurrence a(n) -2*a(n-1) +50*(n-1)*a(n-2)=0. [DLMF] - R. J. Mathar, Feb 16 2014
a(n) = (-1)^floor(n/2)*2^ceiling(n/2)*A237987(n). - M. F. Hasler, Feb 16 2014
From G. C. Greubel, Jun 09 2018: (Start)
a(n) = 5^n * Hermite(n,1/5).
E.g.f.: exp(2*x-25*x^2).
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*n!*(2/5)^(n-2k)/(k!*(n-2k)!). (End)

A234359 a(n) = |{2 < k < n-2: 5^{phi(k)} + 5^{phi(n-k)/2} - 1 is prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 1, 2, 4, 2, 4, 4, 3, 4, 3, 6, 5, 4, 6, 7, 8, 6, 7, 11, 7, 10, 9, 9, 7, 10, 11, 8, 7, 11, 10, 9, 6, 11, 15, 4, 14, 5, 14, 11, 13, 9, 13, 6, 12, 10, 12, 11, 10, 10, 13, 9, 7, 11, 7, 11, 4, 11, 9, 10, 6, 11, 8, 4, 10, 12, 13, 9, 7, 9, 6, 12, 10

Offset: 1

Zhi-Wei Sun, Dec 24 2013

nonn

Conjecture: For any integer a > 1, there is a positive integer N(a) such that if n > N(a) then a^{phi(k)} + a^{phi(n-k)/2} - 1 is prime for some 2 < k < n-2. Moreover, we may take N(2) = N(3) = ... = N(6) = N(8) = 5 and N(7) = 17.

Clearly, this conjecture implies that for each a = 2, 3, ... there are infinitely many primes of the form a^{2*k} + a^m - 1, where k and m are positive integers.

			a(6) = 1 since 5^{phi(3)} + 5^{phi(3)/2} - 1 = 29 is prime.
a(11) = 2 since 5^{phi(4)} + 5^{phi(7)/2} - 1 = 149 and 5^{phi(7)} + 5^{phi(4)/2} - 1 = 15629 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2500

Cf. A000010, A000040, A000351, A234309, A234310, A234337, A234344, A234346, A234347

f[n_,k_]:=5^(EulerPhi[k])+5^(EulerPhi[n-k]/2)-1
a[n_]:=Sum[If[PrimeQ[f[n,k]],1,0],{k,3,n-3}]
Table[a[n],{n,1,100}]

A275069 Number A(n,k) of set partitions of [n] such that i-j is a multiple of k for all i,j belonging to the same block; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 5, 1, 1, 1, 1, 2, 15, 1, 1, 1, 1, 1, 4, 52, 1, 1, 1, 1, 1, 2, 10, 203, 1, 1, 1, 1, 1, 1, 4, 25, 877, 1, 1, 1, 1, 1, 1, 2, 8, 75, 4140, 1, 1, 1, 1, 1, 1, 1, 4, 20, 225, 21147, 1, 1, 1, 1, 1, 1, 1, 2, 8, 50, 780, 115975, 1

Offset: 0

Alois P. Heinz, Jul 15 2016

			A(5,0) = 1: 1|2|3|4|5.
A(5,1) = 52 = A000110(5).
A(5,2) = 10: 135|24, 13|24|5, 135|2|4, 13|2|4|5, 15|24|3, 1|24|35, 1|24|3|5, 15|2|3|4, 1|2|35|4, 1|2|3|4|5.
A(5,3) = 4: 14|25|3, 14|2|3|5, 1|25|3|4, 1|2|3|4|5.
A(5,4) = 2: 15|2|3|4, 1|2|3|4|5.
Square array A(n,k) begins:
  1,      1,    1,   1,   1,  1,  1, 1, 1, 1, 1, ...
  1,      1,    1,   1,   1,  1,  1, 1, 1, 1, 1, ...
  1,      2,    1,   1,   1,  1,  1, 1, 1, 1, 1, ...
  1,      5,    2,   1,   1,  1,  1, 1, 1, 1, 1, ...
  1,     15,    4,   2,   1,  1,  1, 1, 1, 1, 1, ...
  1,     52,   10,   4,   2,  1,  1, 1, 1, 1, 1, ...
  1,    203,   25,   8,   4,  2,  1, 1, 1, 1, 1, ...
  1,    877,   75,  20,   8,  4,  2, 1, 1, 1, 1, ...
  1,   4140,  225,  50,  16,  8,  4, 2, 1, 1, 1, ...
  1,  21147,  780, 125,  40, 16,  8, 4, 2, 1, 1, ...
  1, 115975, 2704, 375, 100, 32, 16, 8, 4, 2, 1, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Wikipedia, Partition of a set

Columns k=0-10 give: A000012, A000110, A124419, A275070, A275071, A275072, A275073, A275074, A275075, A275076, A275077.

A(k*n,n) for k=1-4 gives: A000012, A000079, A000351, A001024.

with(combinat):
A:= (n, k)-> mul(bell(floor((n+i)/k)), i=0..k-1):
seq(seq(A(n, d-n), n=0..d), d=0..14);

A[n_, k_] := Product[BellB[Floor[(n+i)/k]], {i, 0, k-1}]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 17 2017, translated from Maple *)

A(n,k) = Product_{i=0..k-1} A000110(floor((n+i)/k)).

A006496 Imaginary part of (1+2i)^n.

Original entry on oeis.org

0, 2, 4, -2, -24, -38, 44, 278, 336, -718, -3116, -2642, 10296, 33802, 16124, -136762, -354144, -24478, 1721764, 3565918, -1476984, -20783558, -34182196, 35553398, 242017776, 306268562, -597551756, -2726446322, -2465133864, 8701963882, 29729597084, 15949374758

Offset: 0

N. J. A. Sloane

The absolute values of these numbers are the even numbers x such that x^2 + y^2 = 5^n with x and y coprime. See A098122. - T. D. Noe, Apr 14 2011

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
George Berzsenyi, Gaussian Fibonacci numbers, Fib. Quart., Vol. 15, No. 3 (1977), pp. 233-236.
Index entries for Gaussian integers and primes.
Index entries for linear recurrences with constant coefficients, signature (2,-5).

Cf. A000351, A001622, A006495, A098122, A045873.

I:=[0,2]; [n le 2 select I[n] else 2*Self(n-1)-5*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Dec 21 2011

LinearRecurrence[{2,-5},{0,2},30] (* Vincenzo Librandi, Dec 21 2011 *)

a(n)=([1, -2; 2, 1]^n)[1,2] \\ Charles R Greathouse IV, Dec 22 2011

a(n) = 2*a(n-1) - 5*a(n-2); a(0)=0, a(1)=2. - T. D. Noe, Nov 09 2006
a(n) = - [M^n]1,2, where M = [1, -2; 2, 1]. - _Simone Severini, Apr 25 2007
A000351(n) = A006495(n)^2 + a(n)^2. - Fabrice Baubet, May 28 2007
From R. J. Mathar, Apr 06 2008: (Start)
O.g.f.: 2*x/(1 - 2*x + 5*x^2).
a(n) = 2*A045873(n). (End)
E.g.f.: exp(x)*sin(2*x). - Sergei N. Gladkovskii, Jul 22 2012
a(n)/A006495(n) = -tan(2*n*arctan(phi)), where phi is the golden ratio (A001622). - Amiram Eldar, Jan 13 2022

Signs from Christian G. Bower, Nov 15 1998
Corrected by T. D. Noe, Nov 09 2006
More terms from R. J. Mathar, Apr 06 2008

A013710 a(n) = 5^(2*n + 1).

Original entry on oeis.org

5, 125, 3125, 78125, 1953125, 48828125, 1220703125, 30517578125, 762939453125, 19073486328125, 476837158203125, 11920928955078125, 298023223876953125, 7450580596923828125, 186264514923095703125, 4656612873077392578125, 116415321826934814453125, 2910383045673370361328125

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (25).

Cf. A000351, A005408, A009969.

[5^(2*n+1): n in [0..20]]; // Vincenzo Librandi, May 26 2011

NestList[25#&,5,20] (* Harvey P. Dale, Aug 08 2013 *)

a(n)=5^(2*n+1) \\ Charles R Greathouse IV, Jul 11 2016

From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 25*a(n-1); a(0)=5.
G.f.: 5/(1-25*x). (End)
E.g.f.: 5*exp(25*x). - Mohammad K. Azarian, Dec 23 2008
a(n) = 5*A009969(n) = A000351(A005408(n)). - Elmo R. Oliveira, Aug 26 2024

cp's OEIS Frontend

A223233 T(n,k)=Rolling icosahedron footprints: number of nXk 0..11 arrays starting with 0 where 0..11 label vertices of an icosahedron and every array movement to a horizontal, diagonal or antidiagonal neighbor moves along an icosahedral edge.

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A303934 Number of ways to write 2*n as p + 2^k + 5^m with p prime and 2^k + 5^m squarefree, where k and m are nonnegative integers.

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A339685 a(n) = Sum_{d|n} 5^(d-1).

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Magma

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PARI

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A075500 Stirling2 triangle with scaled diagonals (powers of 5).

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Maple

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A152187 a(n) = 3*a(n-1) + 5*a(n-2), with a(0)=1, a(1)=5.

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A158960 Numerator of Hermite(n, 1/5).

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Magma

Maple

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PARI

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A234359 a(n) = |{2 < k < n-2: 5^{phi(k)} + 5^{phi(n-k)/2} - 1 is prime}|, where phi(.) is Euler's totient function.

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A275069 Number A(n,k) of set partitions of [n] such that i-j is a multiple of k for all i,j belonging to the same block; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A152187 a(n) = 3a(n-1) + 5a(n-2), with a(0)=1, a(1)=5.