A001019 -id:A001019 - cp's OEIS frontend

A346460 Square array read by downward antidiagonals in which row n lists all numbers k for which all positive integers cannot be colored with two colors without any positive integer x being the same color as nx or kx (for n >= 2).

4, 16, 9, 64, 81, 2, 256, 729, 8, 25, 1024, 6561, 16, 625, 36, 4096, 59049, 32, 15625, 1296, 49, 16384, 531441, 128, 390625, 46656, 2401, 4, 65536, 4782969, 256, 9765625, 1679616, 117649, 16, 3, 262144, 43046721, 512, 244140625, 60466176, 5764801, 64, 27, 100

Offset: 2

M. Eren Kesim, Aug 25 2021

Row n lists all positive integers k for which there exists at least one pair of positive integers (x, y) such that n^x = k^y and x+y is odd.
If n is an element of A007916, then row n lists all perfect powers of n^2.
A positive integer k is in row n if and only if there exists a positive integer x for which A052410(n)^x = k and A007814(A052409(n)) != A007814(x).

			Table begins:
     4,    16,      64,       256,        1024,          4096,           16384, ...
     9,    81,     729,      6561,       59049,        531441,         4782969, ...
     2,     8,      16,        32,         128,           256,             512, ...
    25,   625,   15625,    390625,     9765625,     244140625,      6103515625, ...
    36,  1296,   46656,   1679616,    60466176,    2176782336,      7836416409, ...
    49,  2401,  117649,   5764801,   282475249,   13841287201,    678223072849, ...
     4,    16,      64,       256,        1024,          4096,           16384, ...
     3,    27,      81,       243,        2187,          6561,           19683, ...
   100, 10000, 1000000, 100000000, 10000000000, 1000000000000, 100000000000000, ...

M. Eren Kesim, First Python program for A346460
M. Eren Kesim, Second Python program for A346460

First few rows are A000302, A001019, A346461, A009969, A009980, and A087752.

Cf. A007814, A007916, A013708-A013715, A052409, A052410, A098608, A346459.

Python
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# See links.
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A013615 Triangle of coefficients in expansion of (1+8x)^n.

Original entry on oeis.org

1, 1, 8, 1, 16, 64, 1, 24, 192, 512, 1, 32, 384, 2048, 4096, 1, 40, 640, 5120, 20480, 32768, 1, 48, 960, 10240, 61440, 196608, 262144, 1, 56, 1344, 17920, 143360, 688128, 1835008, 2097152, 1, 64, 1792, 28672, 286720, 1835008, 7340032, 16777216, 16777216

Offset: 0

N. J. A. Sloane

T(n,k) equals the number of n-length words on {0,1,...,8} having n-k zeros. - Milan Janjic, Jul 24 2015

T:= n-> (p-> seq(coeff(p, x, k), k=0..n))((1+8*x)^n):
seq(T(n), n=0..10);  # Alois P. Heinz, Jul 25 2015

G.f.: 1 / [1 - x(1+8y)].

T(n,k) = 8^k*C(n,k) = Sum_{i=n-k..n} C(i,n-k)*C(n,i)*7^(n-i). Row sums are 9^n = A001019. - Mircea Merca, Apr 28 2012

A013791 a(n) = 9^(4*n + 3).

Original entry on oeis.org

729, 4782969, 31381059609, 205891132094649, 1350851717672992089, 8862938119652501095929, 58149737003040059690390169, 381520424476945831628649898809, 2503155504993241601315571986085849, 16423203268260658146231467800709255289

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 (6561).

Subsequence of A001019.

[9^(4*n+3): n in [0..15]]; // Vincenzo Librandi, Jun 28 2011

9^(4*Range[0,20]+3) (* or *) NestList[6561#&,729,20] (* Harvey P. Dale, Jul 30 2021 *)

a(n)=9^(4*n+3) \\ Charles R Greathouse IV, Jul 11 2016

From Philippe Deléham, Dec 07 2008: (Start)
a(n) = 6561*a(n-1); a(0)=729.
G.f.: 729/(1-6561*x).
a(n) = 81*A013790(n). (End)

A014393 Final 2 digits of 9^n.

Original entry on oeis.org

1, 9, 81, 29, 61, 49, 41, 69, 21, 89, 1, 9, 81, 29, 61, 49, 41, 69, 21, 89, 1, 9, 81, 29, 61, 49, 41, 69, 21, 89, 1, 9, 81, 29, 61, 49, 41, 69, 21, 89, 1, 9, 81, 29, 61, 49, 41, 69, 21, 89, 1, 9, 81, 29, 61, 49, 41, 69, 21, 89

Offset: 0

N. J. A. Sloane

Period is 10, i.e., a(n+10) = a(n). - Martin Renner, Jun 11 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for sequences related to final digits of numbers

Cf. A001019 (9^n), A010690 (final digit of 9^n).

[Modexp(9, n, 100): n in [0..110]]; // Vincenzo Librandi, Aug 16 2016

seq(9^n mod 100, n=0..80); # Martin Renner, Jun 11 2020

Flatten[Prepend[FromDigits[Take[IntegerDigits[#],-2]]&/@(9^Range[2,60]),{1,9}]] (* Harvey P. Dale, Jan 22 2011 *)
PowerMod[9, Range[0, 80], 100] (* Vincenzo Librandi, Aug 16 2016 *)

a(n) = lift(Mod(9, 100)^n); \\ Michel Marcus, Aug 16 2016

a(n) = 9^n mod 100. - Martin Renner, Jun 11 2020

A024103 a(n) = 9^n - n^2.

Original entry on oeis.org

1, 8, 77, 720, 6545, 59024, 531405, 4782920, 43046657, 387420408, 3486784301, 31381059488, 282429536337, 2541865828160, 22876792454765, 205891132094424, 1853020188851585, 16677181699666280, 150094635296998797

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (12,-30,28,-9).

Cf. A000290, A001019.

Cf. similar sequences listed in A024025.

[9^n-n^2: n in [0..25]]; // Vincenzo Librandi, Jul 06 2011

Table[9^n - n^2, {n, 0, 25}] (* or *) CoefficientList[Series[(1 - 4 x + 11 x^2 + 8 x^3)/((1 - 9 x) (1 - x)^3), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 06 2014 *)

G.f.: (1-4*x+11*x^2+8*x^3)/((1-9*x)*(1-x)^3). - Vincenzo Librandi, Oct 06 2014
a(n) = 12*a(n-1) -30*a(n-2) +28*a(n-3) -9*a(n-4) for n>3. - Vincenzo Librandi, Oct 06 2014
a(n) = A001019(n) - A000290(n). - Michel Marcus, Oct 06 2014

A024105 a(n) = 9^n - n^4.

Original entry on oeis.org

1, 8, 65, 648, 6305, 58424, 530145, 4780568, 43042625, 387413928, 3486774401, 31381044968, 282429515745, 2541865799768, 22876792416545, 205891132044024, 1853020188786305, 16677181699583048, 150094635296894145

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (14,-55,100,-95,46,-9).

Cf. A000583, A001019.

[9^n-n^4: n in [0..25]]; // Vincenzo Librandi, Jul 06 2011

Table[9^n-n^4,{n,0,25}] (* or *) LinearRecurrence[{14,-55,100,-95,46,-9},{1,8,65,648,6305,58424},30] (* Harvey P. Dale, Oct 07 2013 *)

a(n) = 14*a(n-1) - 55*a(n-2) + 100*a(n-3) - 95*a(n-4) + 46*a(n-5) - 9*a(n-6); a(0)=1, a(1)=8, a(2)=65, a(3)=648, a(4)=6305, a(5)=58424. - Harvey P. Dale, Oct 07 2013

G.f.: (1 - 6*x + 8*x^2 + 78*x^3 + 103*x^4 + 8*x^5)/((1 - x)^5*(1 - 9*x)). - Stefano Spezia, Aug 01 2022

A024108 a(n) = 9^n-n^7.

Original entry on oeis.org

1, 8, -47, -1458, -9823, -19076, 251505, 3959426, 40949569, 382637520, 3476784401, 31361572438, 282393704673, 2541803079812, 22876687041457, 205890961235274, 1853019920416385, 16677181289327896, 150094634684779089, 1350851716779120350

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (17,-100,308,-574,686,-532,260,-73,9).

[9^n-n^7: n in [0..25]]; // Vincenzo Librandi, Jul 06 2011

Table[9^n - n^7, {n, 0, 19}] (* Michael De Vlieger, Mar 20 2015 *)
LinearRecurrence[{17,-100,308,-574,686,-532,260,-73,9},{1,8,-47,-1458,-9823,-19076,251505,3959426,40949569},20] (* Harvey P. Dale, Mar 23 2018 *)

Vec(-(10*x^8+1071*x^7+10627*x^6+20497*x^5+8373*x^4-167*x^3-83*x^2-9*x+1)/((x-1)^8*(9*x-1)) + O(x^100)) \\ Colin Barker, Mar 20 2015

G.f.: -(10*x^8+1071*x^7+10627*x^6+20497*x^5+8373*x^4-167*x^3-83*x^2-9*x+1) / ((x-1)^8*(9*x-1)). - Colin Barker, Mar 20 2015

a(n) = A001019(n) - A001015(n). - Michel Marcus, Mar 20 2015

A038488 Sums of 3 distinct powers of 9.

Original entry on oeis.org

91, 739, 811, 819, 6571, 6643, 6651, 7291, 7299, 7371, 59059, 59131, 59139, 59779, 59787, 59859, 65611, 65619, 65691, 66339, 531451, 531523, 531531, 532171, 532179, 532251, 538003, 538011, 538083, 538731, 590491, 590499, 590571, 591219, 597051, 4782979, 4783051

Offset: 1

Olivier Gérard

Amiram Eldar, Table of n, a(n) for n = 1..10000

Base-9 interpretation of A038445.

Cf. A001019, A038487, A038489.

Sort[Plus @@@ Subsets[9^Range[0, 6], {3}]] (* Amiram Eldar, Jul 14 2022 *)

from math import isqrt, comb
from sympy import integer_nthroot
def A038488(n): return 9**((r:=n-1-comb((m:=integer_nthroot(6*n,3)[0])+(t:=(n>comb(m+2,3)))+1,3))-comb((k:=isqrt(b:=r+1<<1))+(b>k*(k+1)),2))+9**((a:=isqrt(s:=n-comb(m-(t^1)+2,3)<<1))+((s<<2)>(a<<2)*(a+1)+1))+9**(m+t+1) # Chai Wah Wu, Apr 05 2025

Offset corrected by Amiram Eldar, Jul 14 2022

A038489 Sums of 4 distinct powers of 9.

Original entry on oeis.org

820, 6652, 7300, 7372, 7380, 59140, 59788, 59860, 59868, 65620, 65692, 65700, 66340, 66348, 66420, 531532, 532180, 532252, 532260, 538012, 538084, 538092, 538732, 538740, 538812, 590500, 590572, 590580, 591220, 591228, 591300, 597052, 597060, 597132, 597780, 4783060

Offset: 1

Olivier Gérard

Amiram Eldar, Table of n, a(n) for n = 1..10000

Base-9 interpretation of A038446.

Cf. A001019, A038487, A038488.

Sort[Plus @@@ Subsets[9^Range[0, 6], {4}]] (* Amiram Eldar, Jul 14 2022 *)

from itertools import islice
def A038489_gen(): # generator of terms
    yield int(bin(n:=15)[2:],9)
    while True: yield int(bin((n:=n^((a:=-n&n+1)|(a>>1)) if n&1 else ((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b))[2:],9)
A038489_list = list(islice(A038489_gen(),30)) # Chai Wah Wu, Apr 05 2025

Offset corrected by Amiram Eldar, Jul 14 2022

A055995 a(n) = 64*9^(n-2), a(0)=1, a(1)=7.

Original entry on oeis.org

1, 7, 64, 576, 5184, 46656, 419904, 3779136, 34012224, 306110016, 2754990144, 24794911296, 223154201664, 2008387814976, 18075490334784, 162679413013056, 1464114717117504, 13177032454057536, 118593292086517824

Offset: 0

Barry E. Williams, Jun 04 2000

For n>=2, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4,5,6,7,8,9} such that for fixed, different x_1, x_2 in {1,2,...,n} and fixed y_1, y_2 in {1,2,3,4,5,6,7,8,9} we have f(x_1)<>y_1 and f(x_2)<> y_2. - Milan Janjic, Apr 19 2007

a(n) is the number of generalized compositions of n when there are 8*i-1 different types of i, (i=1,2,...). - Milan Janjic, Aug 26 2010

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (9)

Second differences of 9^n (A001019). Cf. A055275.

a(n) = 9a(n-1) + ((-1)^n)*C(2, 2-n).
G.f.: (1-x)^2/(1-9x).
a(n) = Sum_{k, 0<=k<=n} A201780(n,k)*7^k. - Philippe Deléham, Dec 05 2011

cp's OEIS Frontend

A346460 Square array read by downward antidiagonals in which row n lists all numbers k for which all positive integers cannot be colored with two colors without any positive integer x being the same color as n*x or k*x (for n >= 2).

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Python

A013615 Triangle of coefficients in expansion of (1+8x)^n.

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Maple

Formula

A013791 a(n) = 9^(4*n + 3).

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Magma

Mathematica

PARI

Formula

A014393 Final 2 digits of 9^n.

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Magma

Maple

Mathematica

PARI

Formula

A024103 a(n) = 9^n - n^2.

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Magma

Mathematica

Formula

A024105 a(n) = 9^n - n^4.

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Magma

Mathematica

Formula

A024108 a(n) = 9^n-n^7.

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Magma

Mathematica

PARI

Formula

A038488 Sums of 3 distinct powers of 9.

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Author

Keywords

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Crossrefs

Programs

A346460 Square array read by downward antidiagonals in which row n lists all numbers k for which all positive integers cannot be colored with two colors without any positive integer x being the same color as nx or kx (for n >= 2).