A054605
a(n) = Sum_{d|6} phi(d)*n^(6/d).
Original entry on oeis.org
0, 6, 84, 780, 4200, 15810, 46956, 118104, 262800, 532350, 1001220, 1773156, 2988024, 4829370, 7532700, 11394480, 16781856, 24143094, 34018740, 47053500, 64008840, 85776306, 113391564, 148049160, 191118000, 244157550, 308934756
Offset: 0
-
Table[n^6+n^3+2n^2+2n,{n,0,30}] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,6,84,780,4200,15810,46956},30] (* Harvey P. Dale, Mar 11 2023 *)
A054606
a(n) = Sum_{d|7} phi(d)*n^(7/d).
Original entry on oeis.org
0, 7, 140, 2205, 16408, 78155, 279972, 823585, 2097200, 4783023, 10000060, 19487237, 35831880, 62748595, 105413588, 170859465, 268435552, 410338775, 612220140, 893871853, 1280000120, 1801088667, 2494358020, 3404825585
Offset: 0
- Seiichi Manyama, Table of n, a(n) for n = 0..10000
- Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).
A054607
a(n) = Sum_{d|8} phi(d)*n^(8/d).
Original entry on oeis.org
0, 8, 288, 6672, 65840, 391320, 1681008, 5767328, 16781472, 43053480, 100010240, 214373808, 430002768, 815759672, 1475827920, 2562941760, 4295033408, 6975841608, 11020066272, 16983694160, 25600160880, 37823054808, 54876108848
Offset: 0
- Seiichi Manyama, Table of n, a(n) for n = 0..10000
- Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).
A054608
a(n) = Sum_{d|9} phi(d)*n^(9/d).
Original entry on oeis.org
0, 9, 540, 19755, 262296, 1953405, 10078164, 40354335, 134218800, 387422001, 1000002060, 2357950419, 5159783880, 10604503845, 20661052356, 38443366215, 68719485024, 118587886425, 198359302140, 322687711611, 512000016120
Offset: 0
- Seiichi Manyama, Table of n, a(n) for n = 0..10000
- Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
A054609
a(n) = Sum_{d|10} phi(d)*n^(10/d).
Original entry on oeis.org
0, 10, 1080, 59340, 1049680, 9768870, 60474120, 282492280, 1073774880, 3486843810, 10000100440, 25937586180, 61917613680, 137858863870, 289255193640, 576651150960, 1099512677440, 2015995321530, 3570469117560, 6131068735420, 10240003201680, 16679885064150
Offset: 0
- T. D. Noe, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
-
Table[DivisorSum[10, EulerPhi[#] n^(10/#) &], {n, 0, 100}] (* T. D. Noe, Mar 27 2012 *)
A319082
A(n, k) = (1/k)*Sum_{d|k} EulerPhi(d)*n^(k/d) for n >= 0 and k > 0, A(n, 0) = 0, square array read by ascending antidiagonals.
Original entry on oeis.org
0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 3, 3, 1, 0, 0, 4, 6, 4, 1, 0, 0, 5, 10, 11, 6, 1, 0, 0, 6, 15, 24, 24, 8, 1, 0, 0, 7, 21, 45, 70, 51, 14, 1, 0, 0, 8, 28, 76, 165, 208, 130, 20, 1, 0, 0, 9, 36, 119, 336, 629, 700, 315, 36, 1, 0, 0, 10, 45, 176, 616, 1560, 2635, 2344, 834, 60, 1, 0, 0, 11, 55, 249, 1044, 3367, 7826, 11165, 8230, 2195, 108, 1, 0
Offset: 0
Array starts:
[n\k][0 1 2 3 4 5 6 7 8 9 ...]
[0] 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
[1] 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
[2] 0, 2, 3, 4, 6, 8, 14, 20, 36, 60, ...
[3] 0, 3, 6, 11, 24, 51, 130, 315, 834, 2195, ...
[4] 0, 4, 10, 24, 70, 208, 700, 2344, 8230, 29144, ...
[5] 0, 5, 15, 45, 165, 629, 2635, 11165, 48915, 217045, ...
[6] 0, 6, 21, 76, 336, 1560, 7826, 39996, 210126, 1119796, ...
[7] 0, 7, 28, 119, 616, 3367, 19684, 117655, 720916, 4483815, ...
- D. E. Knuth, Generating All Tuples and Permutations. The Art of Computer Programming, Vol. 4, Fascicle 2, Addison-Wesley, 2005.
- Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
- H. Fredricksen and I. J. Kessler, An algorithm for generating necklaces of beads in two colors, Discrete Math. 61 (1986), 181-188.
- H. Fredricksen and J. Maiorana, Necklaces of beads in k colors and k-ary de Bruijn sequences, Discrete Math. 23(3) (1978), 207-210. Reviewed in MR0523071 (80e:05007).
- Peter Luschny, Implementation of the FKM algorithm in SageMath and Julia
- F. Ruskey, C. Savage, and T. M. Y. Wang, Generating necklaces, Journal of Algorithms, 13(3), 1992, 414-430.
- Index entries for sequences related to necklaces
Essentially the same table as
A075195.
A054630(n,k) is a subtriangle for n >= 1 and 1 <= k <= n.
-
with(numtheory):
A := (n, k) -> `if`(k=0, 0, (1/k)*add(phi(d)*n^(k/d), d=divisors(k))):
seq(seq(A(n-k, k), k=0..n), n=0..12);
# Alternatively, row-wise printed as a table:
T := (n, k) -> `if`(k=0, 0, add(n^igcd(i, k), i=1..k)/k):
seq(lprint(seq(T(n, k), k=0..9)), n=0..7);
-
A(n,k)=if(k==0, 0, sumdiv(k,d, eulerphi(d)*n^(k/d))/k) \\ Andrew Howroyd, Jan 05 2024
-
def A319082(n, k):
return 0 if k == 0 else (1/k)*sum(euler_phi(d)*n^(k//d) for d in divisors(k))
for n in (0..7):
print([n], [A319082(n, k) for k in (0..9)])