A000420 -id:A000420 - cp's OEIS frontend

A047853 a(n) = A047848(5, n).

1, 2, 10, 74, 586, 4682, 37450, 299594, 2396746, 19173962, 153391690, 1227133514, 9817068106, 78536544842, 628292358730, 5026338869834, 40210710958666, 321685687669322, 2573485501354570, 20587884010836554, 164703072086692426, 1317624576693539402, 10540996613548315210

Offset: 0

Clark Kimberling

n-th difference of a(n), a(n-1), ..., a(0) is A000420(n-1) for n >= 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (9,-8).

Cf. A000420, A047848, A226308.

[(8^n +6)/7: n in [0..40]]; // G. C. Greubel, Jan 12 2025

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=8*a[n-1]+1 od: seq(a[n]+1, n=0..18); # Zerinvary Lajos, Mar 20 2008

LinearRecurrence[{9, -8}, {1, 2}, 30] (* Harvey P. Dale, Dec 11 2016 *)
(8^Range[0,40] +6)/7 (* G. C. Greubel, Jan 12 2025 *)

def A047853(n): return (pow(8,n) +6)//7
print([A047853(n) for n in range(41)]) # G. C. Greubel, Jan 12 2025

a(n) = (8^n + 6)/7. - Ralf Stephan, Feb 14 2004
From Philippe Deléham, Oct 05 2009: (Start)
a(0)=1, a(1)=2; a(n) = 9*a(n-1) - 8*a(n-2) for n>1.
G.f.: (1 - 7*x)/(1 - 9*x + 8*x^2). (End)
a(n) = 8*a(n-1) - 6 for n>0, a(0)=1. - Vincenzo Librandi, Aug 06 2010
a(n+1) = A226308(3*n). - Philippe Deléham, Feb 24 2014
E.g.f.: exp(x)*(6 + exp(7*x))/7. - Stefano Spezia, Oct 16 2023

More terms from Vladimir Joseph Stephan Orlovsky, Nov 07 2008

A165828 Totally multiplicative sequence with a(p) = 7.

Original entry on oeis.org

1, 7, 7, 49, 7, 49, 7, 343, 49, 49, 7, 343, 7, 49, 49, 2401, 7, 343, 7, 343, 49, 49, 7, 2401, 49, 49, 343, 343, 7, 343, 7, 16807, 49, 49, 49, 2401, 7, 49, 49, 2401, 7, 343, 7, 343, 343, 49, 7, 16807, 49, 343

Offset: 1

Jaroslav Krizek, Sep 28 2009

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000420, A001222.

7^PrimeOmega[Range[100]] (* G. C. Greubel, Apr 09 2016 *)

a(n) = 7^bigomega(n); \\ Altug Alkan, Apr 09 2016

a(n) = A000420(A001222(n)) = 7^bigomega(n) = 7^A001222(n).

Dirichlet g.f.: Product_{p prime} 1 / (1 - 7 * p^(-s)). - Ilya Gutkovskiy, Oct 30 2019

A175169 Numbers k that divide the sum of digits of 2^k.

Original entry on oeis.org

1, 2, 5, 70

Offset: 1

N. J. A. Sloane, Dec 03 2010

No other terms <= 200000. - Harvey P. Dale, Dec 16 2010
No other terms <= 1320000. - Robert G. Wilson v, Dec 18 2010
There are almost certainly no further terms.

Sum of digits of k^n mod n: (k=2) A000079, A001370, A175434, A175169; (k=3) A000244, A004166, A175435, A067862; (k=5) A000351, A066001, A175456; (k=6) A000400, A066002, A175457, A067864; (k=7) A000420, A066003, A175512, A067863; (k=8) A062933; (k=13) A001022, A175527, A175528, A175525; (k=21) A175589; (k=167) A175558, A175559, A175560, A175552.

Select[Range[1000],Divisible[Total[IntegerDigits[2^#]],#]&] (* Harvey P. Dale, Dec 16 2010 *)

is(n)=sumdigits(2^n)%n==0 \\ Charles R Greathouse IV, Sep 06 2016

A175525 Numbers k that divide the sum of digits of 13^k.

Original entry on oeis.org

1, 2, 5, 140, 158, 428, 788, 887, 914, 1814, 1895, 1976, 2579, 2732, 3074, 3299, 3641, 4658, 4874, 5378, 5423, 5504, 6170, 6440, 6944, 8060, 8249, 8915, 9041, 9158, 9725, 9824, 10661, 11291, 13820, 15305, 17051, 17393, 18716, 19589, 20876, 21641, 23756, 24188, 25961, 28409, 30632, 31307, 32387, 33215, 34970, 35240, 36653, 36977, 41558, 43970, 44951, 47444, 51764, 52655, 53375, 53852, 54104, 56831, 57506, 59153, 66479, 68063, 73562, 78485, 79286, 87908, 92093, 102029, 106934, 114854, 116321, 134051, 139397, 184037, 192353, 256469, 281381, 301118, 469004

Offset: 1

T. D. Noe, Dec 03 2010

Almost certainly there are no further terms.
Comments from Donovan Johnson on the computation of this sequence, Dec 05 2010 (Start):
The number of digits of 13^k is approximately 1.114*k, so I defined an array d() that is a little bigger than 1.114 times the maximum k value to be checked. The elements of d() each are the value of a single digit of the decimal expansion of 13^k with d(1) being the least significant digit.
It's easier to see how the program works if I start with k = 2.
For k = 1, d(2) would have been set to 1 and d(1) would have been set to 3.
k = 2:
x = 13*d(1) = 13*3 = 39
y = 39\10 = 3 (integer division)
x-y*10 = 39-30 = 9, d(1) is set to 9
x = 13*d(2)+y = 13*1+3 = 16, y is the carry from previous digit
y = 16\10 = 1
x-y*10 = 16-10 = 6, d(2) is set to 6
x = 13*d(3)+y = 13*0+1 = 1, y is the carry from previous digit
y = 1\10 = 0
x-y*10 = 1-0 = 1, d(3) is set to 1
These steps would of course be inside a loop and that loop would be inside a k loop. A pointer to the most significant digit increases usually by one and sometimes by two for each successive k value checked. The number of steps of the inner loop is the size of the pointer. A scan is done from the first element to the pointer element to get the digit sum.
(End)
No other terms < 3*10^6. - Donovan Johnson, Dec 07 2010

Sum of digits of k^n mod n: (k=2) A000079, A001370, A175434, A175169; (k=3) A000244, A004166, A175435, A067862; (k=5) A000351, A066001, A175456; (k=6) A000400, A066002, A175457, A067864; (k=7) A000420, A066003, A175512, A067863; (k=8) A062933; (k=13) A001022, A175527, A175528, A175525; (k=21) A175589; (k=167) A175558, A175559, A175560, A175552.

Select[Range[1000], Mod[Total[IntegerDigits[13^#]], #] == 0 &]

a(47)-a(79) from N. J. A. Sloane, Dec 04 2010

a(80)-a(85) from Donovan Johnson, Dec 05 2010

A109808 a(n) = 2*7^(n-1).

Original entry on oeis.org

2, 14, 98, 686, 4802, 33614, 235298, 1647086, 11529602, 80707214, 564950498, 3954653486, 27682574402, 193778020814, 1356446145698, 9495123019886, 66465861139202, 465261027974414, 3256827195820898, 22797790370746286, 159584532595224002, 1117091728166568014

Offset: 1

Woong Kook (andrewk(AT)math.uri.edu), Aug 16 2005

Value of Tutte dichromatic polynomial T_G(0,1) where G is the Cartesian product of the paths P_2 and P_n (n>1).
The value of Tutte dichromatic polynomial T_G(0,1) where G is the Cartesian product of the paths P_1 and P_n (n>1) is seen to be 2^(n-1), which is also the number of edge-rooted forests in P_n.
In 1956, Andrzej Schinzel showed that for every n >= 2, a(n) is not a value of Euler's function. - Arkadiusz Wesolowski, Oct 20 2013
Apart from first term 2, these are the numbers that satisfy phi(n) = 3*n/7. - Michel Marcus, Jul 14 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Tanya Khovanova, Recursive Sequences.
W. Kook, Edge-rooted forests and alpha-invariant of cone graphs, Discrete Applied Mathematics, Volume 155, Issue 8, 15 April 2007, Pages 1071-1075.
Mitchell Paukner, Lucy Pepin, Manda Riehl, and Jarred Wieser, Pattern Avoidance in Task-Precedence Posets, arXiv:1511.00080 [math.CO], 2015-2016.
Index entries for linear recurrences with constant coefficients, signature (7).

Cf. A000420 (powers of 7), A005277 (nontotients), A132023.

[2*7^(n-1):n in [1..25]]; // Vincenzo Librandi, Sep 15 2011

Maple
```
a:= n-> 2*7^(n-1): seq(a(n), n=1..30);
```

2*7^Range[0, 40] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)

a(n)=7^n*2/7 \\ Charles R Greathouse IV, Jun 10 2011

a(n) = 2*7^(n-1); a(n) = 7*a(n-1) where a(1) = 2.
G.f.: 2*x/(1 - 7*x). - Philippe Deléham, Nov 23 2008
E.g.f.: 2*(exp(7*x) - 1)/7. - Stefano Spezia, May 29 2021
From Amiram Eldar, May 08 2023: (Start)
Sum_{n>=1} 1/a(n) = 7/12.
Sum_{n>=1} (-1)^(n+1)/a(n) = 7/16.
Product_{n>=1} (1 - 1/a(n)) = A132023. (End)

Name changed by Arkadiusz Wesolowski, Oct 20 2013

A128965 a(n) = (n^3 - n)*7^n.

Original entry on oeis.org

0, 294, 8232, 144060, 2016840, 24706290, 276710448, 2905459704, 29054597040, 279650496510, 2610071300760, 23751648836916, 211605598728888, 1851548988877770, 15951806673408480, 135590356723972080, 1138958996481365472, 9467596658251350486, 77968443067952298120

Offset: 1

Mohammad K. Azarian, Apr 28 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (28,-294,1372,-2401).

Cf. A000420, A007531, A036289, A128796, A140107.

Cf. A128960, A128961, A128962, A128963, A128964, A128967, A128969.

[(n^3 - n)*7^n: n in [1..25]]; // Vincenzo Librandi, Feb 11 2013

LinearRecurrence[{28, -294, 1372, -2401}, {0, 294, 8232, 144060}, 30] (* Vincenzo Librandi, Feb 11 2013 *)
Table[(n^3-n)7^n,{n,20}] (* Harvey P. Dale, May 14 2020 *)

From R. J. Mathar, Dec 19 2008: (Start)
G.f.: 294x^2/(1-7x)^4.
a(n) = 294*A140107(n-2). (End)
a(n) = 28*a(n-1) - 294*a(n-2) + 1372*a(n-3) - 2401*a(n-4). - Vincenzo Librandi, Feb 11 2013
From Amiram Eldar, Oct 02 2022: (Start)
a(n) = A007531(n+1)*A000420(n).
Sum_{n>=2} 1/a(n) = (18/7)*log(7/6) - 11/28.
Sum_{n>=2} (-1)^n/a(n) = (32/7)*log(8/7) - 17/28. (End)

Offset corrected by Mohammad K. Azarian, Nov 20 2008

A175552 Numbers k such that the digit sum of 167^k is divisible by k.

Original entry on oeis.org

1, 2, 5, 7, 22, 490, 724, 778, 868, 994, 1109, 1390, 1415, 1462, 1642, 1739, 1829, 2146, 2362, 3136, 4954, 6437, 6628, 7103, 11200, 12424, 12863, 14242, 14249, 15059, 15203, 16222, 17140, 18353, 19192, 21233, 22853, 24106, 24574, 24833, 26896, 27652, 28253, 30323, 31306, 31594, 32386, 33790, 34985, 36184, 36310, 40673, 42196, 43931, 45911, 45983

Offset: 1

N. J. A. Sloane, Dec 03 2010

From Donovan Johnson, Dec 03 2010: (Start)
To generate the additional terms I used PFGW.exe to get the decimal expansion for each number of the form 167^n (n <= 50000). Then I wrote a program in powerbasic to read the pfgw.out file and get the digit sums.
The digit sum is 10 times the n value for terms a(5) to a(56). (End)
I believe that this sequence is finite. - N. J. A. Sloane, Dec 05 2010

Donovan Johnson, Table of n, a(n) for n = 1..129

Sum of digits of k^n mod n: (k=2) A000079, A001370, A175434, A175169; (k=3) A000244, A004166, A175435, A067862; (k=5) A000351, A066001, A175456; (k=6) A000400, A066002, A175457, A067864; (k=7) A000420, A066003, A175512, A067863; (k=8) A062933; (k=13) A001022, A175527, A175528, A175525; (k=21) A175589; (k=167) A175558, A175559, A175560, A175552.

Select[Range[10000], Mod[Total[IntegerDigits[167^#]], #] == 0 &]

a(25)-a(56) from Donovan Johnson, Dec 03 2010

A212701 Main transitions in systems of n particles with spin 3.

Original entry on oeis.org

6, 84, 882, 8232, 72030, 605052, 4941258, 39530064, 311299254, 2421216420, 18643366434, 142367525496, 1079620401678, 8138676874188, 61040076556410, 455765904954528, 3389758918099302, 25124095510618356, 185639150161791186, 1367867422244777160, 10053825553499112126

Offset: 1

Stanislav Sykora, May 25 2012

Please, refer to the general explanation in A212697.

This sequence is for base b=7 (see formula), corresponding to spin S=(b-1)/2=3.

Stanislav Sykora, Table of n, a(n) for n = 1..100
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (14,-49).

Cf. A001787, A212697, A212698, A212699, A212700, A212702, A212703, A212704 (b = 2, 3, 4, 5, 6, 8, 9, 10).

Cf. A000420, A008588, A027473.

LinearRecurrence[{14,-49},{6,84},20] (* Harvey P. Dale, Aug 02 2016 *)

mtrans(n, b) = n*(b-1)*b^(n-1);
for (n=1, 100, write("b212701.txt", n, " ", mtrans(n, 7)))

Vec(6*x/(7*x-1)^2 + O(x^100)) \\ Colin Barker, Jun 16 2015

a(n) = n*(b-1)*b^(n-1). For this sequence, set b=7.
From Colin Barker, Jun 16 2015: (Start)
a(n) = 14*a(n-1) - 49*a(n-2) for n > 2.
G.f.: 6*x/(7*x-1)^2. (End)
From Elmo R. Oliveira, May 14 2025: (Start)
E.g.f.: 6*x*exp(7*x).
a(n) = 6*A027473(n) = A008588(n)*A000420(n-1). (End)

A319075 Square array T(n,k) read by antidiagonal upwards in which row n lists the n-th powers of primes, hence column k lists the powers of the k-th prime, n >= 0, k >= 1.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 8, 9, 5, 1, 16, 27, 25, 7, 1, 32, 81, 125, 49, 11, 1, 64, 243, 625, 343, 121, 13, 1, 128, 729, 3125, 2401, 1331, 169, 17, 1, 256, 2187, 15625, 16807, 14641, 2197, 289, 19, 1, 512, 6561, 78125, 117649, 161051, 28561, 4913, 361, 23, 1, 1024, 19683, 390625, 823543, 1771561, 371293

Offset: 0

Omar E. Pol, Sep 09 2018

If n = p - 1 where p is prime, then row n lists the numbers with p divisors.

The partial sums of column k give the column k of A319076.

			The corner of the square array is as follows:
         A000079 A000244 A000351  A000420    A001020    A001022     A001026
A000012        1,      1,      1,       1,         1,         1,          1, ...
A000040        2,      3,      5,       7,        11,        13,         17, ...
A001248        4,      9,     25,      49,       121,       169,        289, ...
A030078        8,     27,    125,     343,      1331,      2197,       4913, ...
A030514       16,     81,    625,    2401,     14641,     28561,      83521, ...
A050997       32,    243,   3125,   16807,    161051,    371293,    1419857, ...
A030516       64,    729,  15625,  117649,   1771561,   4826809,   24137569, ...
A092759      128,   2187,  78125,  823543,  19487171,  62748517,  410338673, ...
A179645      256,   6561, 390625, 5764801, 214358881, 815730721, 6975757441, ...
...

Rows 0-13: A000012, A000040, A001248, A030078, A030514, A050997, A030516, A092759, A179645, A179665, A030629, A079395, A030631, A138031.
Other rows n: A030635 (n=16), A030637 (n=18), A137486 (n=22), A137492 (n=28), A139571 (n=30), A139572 (n=36), A139573 (n=40), A139574 (n=42), A139575 (n=46), A173533 (n=52), A183062 (n=58), A183085 (n=60), A261700 (n=100).
Columns 1-15: A000079, A000244, A000351, A000420, A001020, A001022, A001026, A001029, A009967, A009973, A009975, A009981, A009985, A009987, A009991.
Main diagonal gives A093360.
Second diagonal gives A062457.
Third diagonal gives A197987.
Removing the 1's we have A182944/ A182945.
Cf. A006093, A319074, A319076.

PARI
```
T(n, k) = prime(k)^n;
```

T(n,k) = A000040(k)^n, n >= 0, k >= 1.

A013613 Triangle of coefficients in expansion of (1+6x)^n.

Original entry on oeis.org

1, 1, 6, 1, 12, 36, 1, 18, 108, 216, 1, 24, 216, 864, 1296, 1, 30, 360, 2160, 6480, 7776, 1, 36, 540, 4320, 19440, 46656, 46656, 1, 42, 756, 7560, 45360, 163296, 326592, 279936, 1, 48, 1008, 12096, 90720, 435456, 1306368, 2239488, 1679616

Offset: 0

N. J. A. Sloane

T(n,k) equals the number of n-length words on {0,1,...,6} having n-k zeros. - Milan Janjic, Jul 24 2015

			Triangle begins:
1;
1, 6;
1, 12, 36;
1, 18, 108, 216;
1, 24, 216, 864, 1296;
...

Michael De Vlieger and Reinhard Zumkeller, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened, rows 0..125 from Reinhard Zumkeller)
Ömür Deveci and Anthony G. Shannon, Some aspects of Neyman triangles and Delannoy arrays, Mathematica Montisnigri (2021) Vol. L, 36-43.

Cf. A038255 (mirrored).

import Data.List (inits)
a013613 n k = a013613_tabl !! n !! k
a013613_row n = a013613_tabl !! n
a013613_tabl = zipWith (zipWith (*))
               (tail $ inits a000400_list) a007318_tabl
-- Reinhard Zumkeller, Nov 21 2013

G.f.: 1 / (1 - x(1+6y)).
T(n,k) = 6^k*C(n,k) = Sum_{i=n-k..n} C(i,n-k)*C(n,i)*5^(n-i). Row sums are 7^n = A000420. - Mircea Merca, Apr 28 2012
T(n,k) = A007318(n,k)*A000400(k), 0 <= k <= n. - Reinhard Zumkeller, Nov 21 2013

cp's OEIS Frontend

A047853 a(n) = A047848(5, n).

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A165828 Totally multiplicative sequence with a(p) = 7.

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A175169 Numbers k that divide the sum of digits of 2^k.

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A175525 Numbers k that divide the sum of digits of 13^k.

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A109808 a(n) = 2*7^(n-1).

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A128965 a(n) = (n^3 - n)*7^n.

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A175552 Numbers k such that the digit sum of 167^k is divisible by k.

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Mathematica

Extensions

A212701 Main transitions in systems of n particles with spin 3.

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