A009968 -id:A009968 - cp's OEIS frontend

A067426 Sixth column of triangle A067425.

1, 9, 216, 5184, 124416, 2985984, 71663616, 1719926784, 41278242816, 990677827584, 23776267862016, 570630428688384, 13695130288521216, 328683126924509184, 7888395046188220416, 189321481108517289984

Offset: 0

Wolfdieter Lang, Jan 25 2002

The fifth column of A067425 gives [1, 8, 160, 3200, 64000, 1280000, 25600000, ...].

Vincenzo Librandi, Table of n, a(n) for n = 0..700
Index entries for linear recurrences with constant coefficients, signature (24).

Cf. A009968 (powers of 24), A067404 (fourth column), A067425, A067427 (seventh column).

[Ceiling(9*(4*6)^(n-1)): n in [0..20]]; // Vincenzo Librandi, Oct 02 2011

Table[Ceiling[9(24)^(n-1)],{n,0,15}] (* Paul F. Marrero Romero, Oct 11 2024 *)

a(n) = A067425(n+5, 5).
a(n) = 9*(4*6)^(n-1), n >= 1, a(0)=1.
G.f.: (1-15*x)/(1-24*x).
E.g.f.: (3*exp(24*x) + 5)/8. - Stefano Spezia, Oct 15 2024

A100403 Digital root of 6^n.

Original entry on oeis.org

1, 6, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 0

Cino Hilliard, Dec 31 2004

Also the digital root of k^n for any k == 6 (mod 9). - Timothy L. Tiffin, Dec 02 2023

			For n=8, the digits of 6^8 = 1679616 sum to 36, whose digits sum to 9. So, a(8) = 9. - _Timothy L. Tiffin_, Dec 01 2023

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000400, A001024, A007953, A009968, A009977, A009986, A010888, A100401, A159991.

PadRight[{1, 6}, 100, 9] (* Timothy L. Tiffin, Dec 03 2023 *)

a(n) = if( n<2, [1,6][n+1], 9); \\ Joerg Arndt, Dec 03 2023

From Timothy L. Tiffin, Dec 01 2023: (Start)
a(n) = 9 for n >= 2.
G.f.: (1+5x+3x^2)/(1-x).
a(n) = A100401(n) for n <> 1.
a(n) = A010888(A000400(n)) = A010888(A001024(n)) = A010888(A009968(n)) = A010888(A009977(n)) = A010888(A009986(n)) = A010888(A159991(n)). (End)
E.g.f.: 9*exp(x) - 3*x - 8. - Elmo R. Oliveira, Aug 09 2024
a(n) = A007953(6*a(n-1)) = A010888(6*a(n-1)). - Stefano Spezia, Mar 20 2025

A218727 a(n) = (24^n - 1)/23.

Original entry on oeis.org

0, 1, 25, 601, 14425, 346201, 8308825, 199411801, 4785883225, 114861197401, 2756668737625, 66160049703001, 1587841192872025, 38108188628928601, 914596527094286425, 21950316650262874201, 526807599606308980825, 12643382390551415539801, 303441177373233972955225

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 24 (A009968); q-integers for q=24: diagonal k=1 in triangle A022188.

Partial sums are in A014913. Also, the sequence is related to A014942 by A014942(n) = n*a(n) - Sum_{i=0..n-1} a(i) for n > 0. [Bruno Berselli, Nov 07 2012]

Vincenzo Librandi, Table of n, a(n) for n = 0..700
Index entries related to partial sums.
Index entries for linear recurrences with constant coefficients, signature (25,-24).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009968, A014942, A022188.

[n le 2 select n-1 else 25*Self(n-1)-24*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{25, -24}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218727(n):=(24^n-1)/23$
makelist(A218727(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218727(n)=24^n\23
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-24*x)).
a(n) = floor(24^n/23).
a(n) = 25*a(n-1) - 24*a(n-2). (End)
E.g.f.: exp(x)*(exp(23*x) - 1)/23. - Elmo R. Oliveira, Aug 29 2024

A100685 Powers of factorials A000142.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 24, 32, 36, 64, 120, 128, 216, 256, 512, 576, 720, 1024, 1296, 2048, 4096, 5040, 7776, 8192, 13824, 14400, 16384, 32768, 40320, 46656, 65536, 131072, 262144, 279936, 331776, 362880, 518400, 524288, 1048576, 1679616, 1728000

Offset: 1

Kyle Schalm and Jonathan Sondow, Dec 08 2004

Subsequence of A001013. Supersequence of A036740 without its first term.

Supersequence also of A046882 and A055209 without their first terms. - Jonathan Sondow and Robert G. Wilson v, Dec 19 2004

			24 = (4!)^1 and 36 = (3!)^2.

G. C. Greubel, Table of n, a(n) for n = 1..9660
Index entries for sequences related to factorial numbers.

Cf. A000142, A001013, A036740, A331373.
Cf. also A046882 and A055209.
Subsequences: A000079, A000400, A009968.

With[{ln = Log[10!]}, Table[With[{f = m!}, Table[f^j, {j, 0, Floor[ln/Log[f]]}]], {m, 2, 10}]] //Flatten //Union

Sum_{n>=1} 1/a(n) = 1 + A331373. - Amiram Eldar, Nov 21 2021

A267144 Octagonal numbers with prime indices.

Original entry on oeis.org

8, 21, 65, 133, 341, 481, 833, 1045, 1541, 2465, 2821, 4033, 4961, 5461, 6533, 8321, 10325, 11041, 13333, 14981, 15841, 18565, 20501, 23585, 28033, 30401, 31621, 34133, 35425, 38081, 48133, 51221, 56033, 57685, 66305, 68101, 73633, 79381, 83333, 89441, 95765

Offset: 1

Ilya Gutkovskiy, Jan 11 2016

Eric Weisstein's World of Mathematics, Octagonal Number
Eric Weisstein's World of Mathematics, Prime Number

Cf. A000040, A000567, A001248, A034953, A116995, A117961.

[NthPrime(n)*(3*NthPrime(n)-2): n in [1..50]]; // Vincenzo Librandi, Jan 12 2016

Table[Prime[n] (3 Prime[n] - 2), {n, 1, 45}]

lista(nn) = forprime(p=2, nn, print1(p*(3*p-2), ", ")); \\ Altug Alkan, Jan 11 2016

from sympy import prime
def a(n): p = prime(n); return p*(3*p-2)
print([a(n) for n in range(1, 42)]) # Michael S. Branicky, Aug 21 2021

a(n) = prime(n)*(3*prime(n) - 2) = A000040(n)*(3*A000040(n) - 2).
a(n) = A000567(A000040(n)).
a(n) = sigma_0(24^(prime(n) - 1)) = A000005(A009968(A000040(n) - 1)).

A013774 a(n) = 24^(3*n + 1).

Original entry on oeis.org

24, 331776, 4586471424, 63403380965376, 876488338465357824, 12116574790945106558976, 167499529910025153071284224, 2315513501476187716057433112576, 32009658644406818986777955348250624

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 (13824).

Subsequence of A009968.

[24^(3*n+1): n in [0..10]]; // Vincenzo Librandi, Jun 27 2011

Table[24^(3*n + 1), {n, 0, 10}] (* Wesley Ivan Hurt, Jan 20 2024 *)

A013910 a(n) = 24^(5*n + 1).

Original entry on oeis.org

24, 191102976, 1521681143169024, 12116574790945106558976, 96479729228174488169059713024, 768231807465763655682670928358014976, 6117141027690268863066571918245810640257024

Offset: 0

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..50
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (7962624).

Cf. A009968.

[24^(5*n+1): n in [0..10]]; // Vincenzo Librandi, May 29 2011

NestList[7962624#&,24,10] (* Harvey P. Dale, Jul 25 2019 *)

a(n) = 7962624*a(n-1), a(0)=24, where 7962624 = 24^5. - Vincenzo Librandi, May 29 2011
G.f.: 24/(1 - 7962624*x). - Stefano Spezia, Sep 09 2018
E.g.f.: 24*exp(7962624*x). - Alejandro J. Becerra Jr., May 18 2021

A013911 a(n) = 24^(5*n + 2).

Original entry on oeis.org

576, 4586471424, 36520347436056576, 290797794982682557415424, 2315513501476187716057433112576, 18437563379178327736384102280592359424, 146811384664566452713597726037899455366168576, 1169003855003308785972158379694803112885582691303424

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..50
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (7962624).

Cf. A009968.

[24^(5*n+2): n in [0..10]]; // Vincenzo Librandi, May 30 2011

a(n)=24^(5*n+2) \\ Charles R Greathouse IV, May 30 2011

a(n) = 7962624*a(n-1), a(0)=576, where 7962624 = 24^5 and 576=24^2. - Vincenzo Librandi, May 30 2011

E.g.f.: 576*exp(7962624*x). - Alejandro J. Becerra Jr., May 18 2021

A013912 a(n) = 24^(5*n + 3).

Original entry on oeis.org

13824, 110075314176, 876488338465357824, 6979147079584381377970176, 55572324035428505185378394701824, 442501521100279865673218454734216626176, 3523473231949594865126345424909586928788045824, 28056092520079410863331801112675274709253984591282176

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..50
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (7962624).

Cf. A009968.

[24^(5*n+3): n in [0..10]]; // Vincenzo Librandi, May 30 2011

24^(5*Range[0,20]+3) (* or *) NestList[7962624#&,13824,20] (* Harvey P. Dale, Mar 02 2013 *)

a(n)=24^(5*n+3) \\ Charles R Greathouse IV, May 30 2011

a(n) = 7962624*a(n-1), a(0)=13824, where 7962624 = 24^5 and 13824=24^3. - Vincenzo Librandi, May 30 2011

E.g.f.: 13824*exp(7962624*x). - Alejandro J. Becerra Jr., May 18 2021

A156959 Numerator of Euler(n, 1/24).

Original entry on oeis.org

1, -11, -23, 3421, 13777, -3946811, -23818823, 9662765101, 77747254177, -40597653164651, -408311621397623, 260621490978697981, 3145444161956190577, -2372776776728647969691, -33409916827040236064423, 29080297194139431470874061, 467960606621237943982982977

Offset: 0

N. J. A. Sloane, Nov 07 2009

T. D. Noe, Table of n, a(n) for n = 0..100

For denominators see A009968.

Numerator[EulerE[Range[0, 20], 1/24]] (* Vincenzo Librandi, Jul 31 2014 *)

cp's OEIS Frontend

A067426 Sixth column of triangle A067425.

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A100403 Digital root of 6^n.

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A218727 a(n) = (24^n - 1)/23.

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A100685 Powers of factorials A000142.

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A267144 Octagonal numbers with prime indices.

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A013774 a(n) = 24^(3*n + 1).

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A013910 a(n) = 24^(5*n + 1).

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A013911 a(n) = 24^(5*n + 2).