A089081 -id:A089081 - cp's OEIS frontend

A003992 Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 9, 8, 1, 0, 1, 5, 16, 27, 16, 1, 0, 1, 6, 25, 64, 81, 32, 1, 0, 1, 7, 36, 125, 256, 243, 64, 1, 0, 1, 8, 49, 216, 625, 1024, 729, 128, 1, 0, 1, 9, 64, 343, 1296, 3125, 4096, 2187, 256, 1, 0, 1, 10, 81, 512, 2401, 7776, 15625, 16384, 6561, 512, 1, 0

Offset: 0

Marc LeBrun

If the array is transposed, T(n,k) is the number of oriented rows of n colors using up to k different colors. The formula would be T(n,k) = [n==0] + [n>0]*k^n. The generating function for column k would be 1/(1-k*x). For T(3,2)=8, the rows are AAA, AAB, ABA, ABB, BAA, BAB, BBA, and BBB. - Robert A. Russell, Nov 08 2018

T(n,k) is the number of multichains of length n from {} to [k] in the Boolean lattice B_k. - Geoffrey Critzer, Apr 03 2020

			Rows begin:
[1, 0,  0,   0,    0,     0,      0,      0, ...],
[1, 1,  1,   1,    1,     1,      1,      1, ...],
[1, 2,  4,   8,   16,    32,     64,    128, ...],
[1, 3,  9,  27,   81,   243,    729,   2187, ...],
[1, 4, 16,  64,  256,  1024,   4096,  16384, ...],
[1, 5, 25, 125,  625,  3125,  15625,  78125, ...],
[1, 6, 36, 216, 1296,  7776,  46656, 279936, ...],
[1, 7, 49, 343, 2401, 16807, 117649, 823543, ...], ...

T. D. Noe, Rows n = 0..50 of triangle, flattened

Rows 0-49 are A000007, A000012, A000079, A000244, A000302, A000351, A000400, A000420, A001018, A001019, A011557, A001020, A001021, A001022, A001023, A001024, A001025, A001026, A001027, A001029, A009964-A009992, A087752.
Columns 0-26 are A000012, A001477, A000290, A000578, A000583, A000584, A001014, A001015, A001016, A001017, A008454, A008455, A008456, A010801-A010813, A089081.
Main diagonal is A000312. Other diagonals include A000169, A007778, A000272, A008788. Antidiagonal sums are in A026898.
Cf. A099555.
Transpose is A004248. See A051128, A095884, A009999 for other versions.
Cf. A277504 (unoriented), A293500 (chiral).

[[(n-k)^k: k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 08 2018

Table[If[k == 0, 1, (n - k)^k], {n, 0, 11}, {k, 0, n}]//Flatten

T(n,k) = (n-k)^k \\ Charles R Greathouse IV, Feb 07 2017

E.g.f.: Sum T(n,k)*x^n*y^k/k! = 1/(1-x*exp(y)). - Paul D. Hanna, Oct 22 2004

E.g.f.: Sum T(n,k)*x^n/n!*y^k/k! = e^(x*e^y). - Franklin T. Adams-Watters, Jun 23 2006

More terms from David W. Wilson

Edited by Paul D. Hanna, Oct 22 2004

A122968 27th powers: a(n) = n^27.

Original entry on oeis.org

0, 1, 134217728, 7625597484987, 18014398509481984, 7450580596923828125, 1023490369077469249536, 65712362363534280139543, 2417851639229258349412352, 58149737003040059690390169

Offset: 0

Franklin T. Adams-Watters, Oct 27 2006

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A010812, A010813, A089081.

lst={}; Do[AppendTo[lst,n^27],{n,0,4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 15 2009 *)
Range[0,10]^27 (* Harvey P. Dale, Dec 17 2011 *)

Totally multiplicative sequence with a(p) = p^27 for prime p. Multiplicative sequence with a(p^e) = p^(27e). - Jaroslav Krizek, Nov 01 2009
From Amiram Eldar, Oct 09 2020: (Start)
Dirichlet g.f.: zeta(s-27).
Sum_{n>=1} 1/a(n) = zeta(27).
Sum_{n>=1} (-1)^(n+1)/a(n) = 67108863*zeta(27)/67108864. (End)

A122969 28th powers: a(n) = n^28.

Original entry on oeis.org

0, 1, 268435456, 22876792454961, 72057594037927936, 37252902984619140625, 6140942214464815497216, 459986536544739960976801, 19342813113834066795298816, 523347633027360537213511521

Offset: 0

Franklin T. Adams-Watters, Oct 27 2006

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A010813, A089081, A122968.

Table[n^28,{n,0,16}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2010 *)
Range[0,16]^28 (* Harvey P. Dale, Dec 09 2012 *)

Totally multiplicative sequence with a(p) = p^28 for prime p. Multiplicative sequence with a(p^e) = p^(28e). - Jaroslav Krizek, Nov 01 2009
From Amiram Eldar, Oct 09 2020: (Start)
Dirichlet g.f.: zeta(s-28).
Sum_{n>=1} 1/a(n) = zeta(28) = 6785560294*Pi^28/564653660170076273671875.
Sum_{n>=1} (-1)^(n+1)/a(n) = 134217727*zeta(28)/134217728 = 65053034220152267*Pi^28/5413323669636552217067520000000. (End)

A122970 29th powers: a(n) = n^29.

Original entry on oeis.org

0, 1, 536870912, 68630377364883, 288230376151711744, 186264514923095703125, 36845653286788892983296, 3219905755813179726837607, 154742504910672534362390528, 4710128697246244834921603689, 100000000000000000000000000000, 1586309297171491574414436704891

Offset: 0

Franklin T. Adams-Watters, Oct 27 2006

The least significant digit of a(n) is the same as the least significant digit of n. - Alonso del Arte, Mar 28 2015

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index to divisibility sequences

Cf. A089081, A122968, A122969.

[n^29: n in [0..10]]; // Vincenzo Librandi, Mar 29 2015

Range[0, 9]^29 (* Alonso del Arte, Mar 28 2015 *)

a(n)=n^29 \\ Charles R Greathouse IV, Jun 28 2015

Completely multiplicative sequence with a(p) = p^29 for prime p. Multiplicative sequence with a(p^e) = p^(29e). - Jaroslav Krizek, Nov 01 2009
From Amiram Eldar, Oct 09 2020: (Start)
Dirichlet g.f.: zeta(s-29).
Sum_{n>=1} 1/a(n) = zeta(29).
Sum_{n>=1} (-1)^(n+1)/a(n) = 268435455*zeta(29)/268435456. (End)

a(10)-a(11) from Michel Marcus, Mar 29 2015

A137490 Numbers with 27 divisors.

Original entry on oeis.org

900, 1764, 2304, 4356, 4900, 6084, 6400, 10404, 11025, 12100, 12544, 12996, 16900, 19044, 23716, 26244, 27225, 28900, 30276, 30976, 33124, 34596, 36100, 38025, 43264, 49284, 52900, 53361, 56644, 60516, 65025, 66564, 70756, 73984, 74529

Offset: 1

R. J. Mathar, Apr 22 2008

nonn

Maple implementation: see A030513.

Numbers of the form p^26 (subset of A089081), p^2*q^2*r^2 (like 900, 1764, 4356, squares of A007304) or p^2*q^8 (like 2304, 6400, subset of the squares of A030628) where p, q and r are distinct primes. - R. J. Mathar, Mar 01 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000005, A030513-A030516, A030626, A030627, A030634-A030638, A005179, A003680, A096932, A061286, A061283.

Select[Range[150000],DivisorSigma[0,#]==27&] (* Vladimir Joseph Stephan Orlovsky, May 06 2011 *)

isA137490(n) = numdiv(n)==27 \\ Michael B. Porter, Apr 10 2010

A000005(a(n)) = 27.

Sum_{n>=1} 1/a(n) = (P(2)^3 + 2*P(6) - 3*P(2)*P(4))/6 + P(2)*P(8) - P(10) + P(26) = 0.00453941..., where P is the prime zeta function. - Amiram Eldar, Jul 03 2022

A112258 Numbers n not divisible by 10 such that the decimal representation of n^26 does not use every nonzero digit.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 17, 23, 29, 39, 61, 81, 95, 119, 164, 242, 5193, 9004, 23432, 246968, 8876708, 32886598, 2141194665

Offset: 1

Klaus Brockhaus, Aug 30 2005

Multiples of 10 are excluded because (10*n)^k uses the same nonzero digits as n^k. - Is the sequence finite?
Similar sequences can be defined for other positive integer exponents. 26 is the smallest exponent such that the corresponding sequence has less than 30 terms < 10^8.
a(29) > 10^11, if it exists. - Chai Wah Wu, Sep 19 2018

			5^26 = 1490116119384765625 uses every digit, so 5 is not in the sequence.
6^26 = 170581728179578208256 does not use digits 3 and 4, so 6 is a term.

Patrick De Geest, The Nine Digits Page
Eric Weisstein's World of Mathematics, Pandigital

Cf. A089081 (26th powers).

{e=26;for(n=1,350000,if(n%10>0,v=vector(9);c=0;k=n^e;while(c<9&&k>0, g=divrem(k,10);k=g[1];if(g[2]>0&&v[g[2]]==0,v[g[2]]=1;c++));if(c<9,print1(n,","))))}

A112258_list = [n for n in range(1,10**6) if n % 10 and len(set(str(n**26))) < 10] # Chai Wah Wu, May 31 2015

a(28) from Lars Blomberg, Sep 25 2011

cp's OEIS Frontend

A003992 Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0.

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A122968 27th powers: a(n) = n^27.

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A122969 28th powers: a(n) = n^28.

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A122970 29th powers: a(n) = n^29.

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A137490 Numbers with 27 divisors.

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A112258 Numbers n not divisible by 10 such that the decimal representation of n^26 does not use every nonzero digit.

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