A010809 -id:A010809 - cp's OEIS frontend

A242038 Integer part of square root of A010809(n) = n^21.

0, 1, 1448, 102275, 2097152, 21836601, 148111277, 747359260, 3037000499, 10460353203, 31622776601, 86024705429, 214488041413, 497055861119, 1082291816449, 2233357359474, 4398046511104, 8312155792152, 15148209527700, 26724698233906

Offset: 0

Karl V. Keller, Jr., Oct 01 2014

nonn

			For n = 4, floor(sqrt(4^21)) = 2097152.

Karl V. Keller, Jr., Table of n, a(n) for n = 0..1000

Cf. A010809 (n^21).

a(n) = sqrtint(n^21) \\ Michel Marcus, Oct 01 2014

from decimal import *
getcontext().prec = 100
for n in range(0,1001): print(int(Decimal(n**21).sqrt()))

a(n) = floor(sqrt(n^21)).

A244003 A(n,k) = k^Fibonacci(n); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 2, 1, 0, 1, 4, 3, 4, 1, 0, 1, 5, 4, 9, 8, 1, 0, 1, 6, 5, 16, 27, 32, 1, 0, 1, 7, 6, 25, 64, 243, 256, 1, 0, 1, 8, 7, 36, 125, 1024, 6561, 8192, 1, 0, 1, 9, 8, 49, 216, 3125, 65536, 1594323, 2097152, 1, 0

Offset: 0

Alois P. Heinz, Jun 17 2014

			Square array A(n,k) begins:
  1, 1,   1,    1,     1,      1,       1, ...
  0, 1,   2,    3,     4,      5,       6, ...
  0, 1,   2,    3,     4,      5,       6, ...
  0, 1,   4,    9,    16,     25,      36, ...
  0, 1,   8,   27,    64,    125,     216, ...
  0, 1,  32,  243,  1024,   3125,    7776, ...
  0, 1, 256, 6561, 65536, 390625, 1679616, ...

Alois P. Heinz, Antidiagonals n = 0..20, flattened

Columns k=0-10 give: A000007, A000012, A000301, A010098, A010099, A214706, A215270, A214887, A215271, A215272, A010100.
Rows n=0, 1+2, 3-8 give: A000012, A001477, A000290, A000578, A000584, A001016, A010801, A010809.
Main diagonal gives: A152915.
Cf. A000045, A103323.

A:= (n, k)-> k^(<<1|1>, <1|0>>^n)[1, 2]:
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[0, 0] = 1; A[n_, k_] := k^Fibonacci[n]; Table[A[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 11 2015 *)

A(n,k) = k^A000045(n).

A(0,k) = 1, A(1,k) = k, A(n,k) = A(n-1,k) * A(n-2,k) for n>=2.

A010810 22nd powers: a(n) = n^22.

Original entry on oeis.org

0, 1, 4194304, 31381059609, 17592186044416, 2384185791015625, 131621703842267136, 3909821048582988049, 73786976294838206464, 984770902183611232881, 10000000000000000000000, 81402749386839761113321

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (23, -253, 1771, -8855, 33649, -100947, 245157, -490314, 817190, -1144066, 1352078, -1352078, 1144066, -817190, 490314, -245157, 100947, -33649, 8855, -1771, 253, -23, 1).

Cf. A010807, A013678, A010809.

[n^22: n in [0..15]]; // Vincenzo Librandi, Jun 19 2011

Table[n^20, {n, 0, 22}] (* Amiram Eldar, Oct 09 2020 *)

a(n) = n^22; \\ Michel Marcus, Feb 27 2018

Totally multiplicative sequence with a(p) = p^22 for prime p. Multiplicative sequence with a(p^e) = p^(22e). - Jaroslav Krizek, Nov 01 2009
From Amiram Eldar, Oct 09 2020: (Start)
Dirichlet g.f.: zeta(s-22).
Sum_{n>=1} 1/a(n) = zeta(22) = 155366*Pi^22/13447856940643125.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2097151*zeta(22)/2097152 = 3324754717*Pi^22/287777551824322560000. (End)

A075821 List of possible last two digits (leading zeros omitted) of perfect powers.

Original entry on oeis.org

0, 1, 3, 4, 7, 8, 9, 11, 12, 13, 16, 17, 19, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 36, 37, 39, 41, 43, 44, 47, 48, 49, 51, 52, 53, 56, 57, 59, 61, 63, 64, 67, 68, 69, 71, 72, 73, 75, 76, 77, 79, 81, 83, 84, 87, 88, 89, 91, 92, 93, 96, 97, 99

Offset: 1

Zak Seidov, Oct 14 2002

An equivalent definition: Numbers equal to the final two digits of their 21st, 41st, 61st, etc. powers. - Henry Bottomley, Nov 25 2004

			With leading zeros, the initial terms are 00, 01, 03, 04, 07, 08, 09. Corresponding smallest perfect powers are 100, 2401, 658503, 2304, 16807, 140608, 2209.
1 (01!) is OK because the perfect power 2401=49^2 ends with 01. 9 (09!) is OK because the perfect power 2209=47^2 ends with 09.
11 is in the sequence since 11^21=7400249944258160101211 and the final two digits are 11.

Cf. A010809, A100990.

Consists of all numbers below 100 except those which are a multiple of 2 but not 4 and those which are a multiple of 5 but not 25. - Henry Bottomley, Nov 25 2004

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 06 2007

A036099 Centered cube numbers: (n+1)^21 + n^21.

Original entry on oeis.org

1, 2097153, 10462450355, 4408506864307, 481235204714229, 22413787798580981, 580482814723661863, 9781917900938059815, 118642361168367135017, 1109418989131512359209, 8400249944258160101211, 53405369853627861567323

Offset: 0

N. J. A. Sloane

After a(0) always has at least 4 prime factors, because a(n) = (2n + 1) * (n^2 + n + 1) * (n^6 + 3n^5 + 9n^4 + 13n^3 + 11n^2 + 5n + 1) * (n^12 + 6n^11 + 63n^10 + 260n^9 + 643n^8 + 1078n^7 + 1275n^6 + 1078n^5 + 650n^4 + 274n^3 + 77n^2 + 13n + 1). [Jonathan Vos Post, Aug 27 2011]

B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Cf. A010809, A036098, A036097.

[(n+1)^21+n^21: n in [0..20]]; // Vincenzo Librandi, Aug 28 2011

Total/@Partition[Range[0,20]^21,2,1] (* Harvey P. Dale, Jul 02 2019 *)

A022537 Nexus numbers (n+1)^21 - n^21.

Original entry on oeis.org

1, 2097151, 10458256051, 4387586157901, 472439111692021, 21460113482174731, 536608913442906151, 8664826172771491801, 100195617094657583401, 890581010868487640791, 6400249944258160101211, 38604869965111541364901, 201059409164080691238301, 924291046880536829143651

Offset: 0

N. J. A. Sloane

nonn

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 54.

G. C. Greubel, Table of n, a(n) for n = 0..10000

Column k=20 of A047969.

Cf. A010809 (n^21).

[(n+1)^21 - n^21: n in [0..20]]; // G. C. Greubel, Feb 27 2018

b:=21: a:=n->(n+1)^b-n^b: seq(a(n),n=0..18); # Muniru A Asiru, Feb 28 2018

Last[#]-First[#]&/@Partition[Range[0,20]^21,2,1] (* Harvey P. Dale, Dec 06 2013 *)

for(n=0,20, print1((n+1)^21 - n^21, ", ")) \\ G. C. Greubel, Feb 27 2018

a(n) = A010809(n+1) - A010809(n). - Michel Marcus, Feb 27 2018

More terms added by G. C. Greubel, Feb 27 2018

A100990 a(n) = n^21 mod 100.

Original entry on oeis.org

0, 1, 52, 3, 4, 25, 56, 7, 8, 9, 0, 11, 12, 13, 64, 75, 16, 17, 68, 19, 0, 21, 72, 23, 24, 25, 76, 27, 28, 29, 0, 31, 32, 33, 84, 75, 36, 37, 88, 39, 0, 41, 92, 43, 44, 25, 96, 47, 48, 49, 0, 51, 52, 53, 4, 75, 56, 57, 8, 59, 0, 61, 12, 63, 64, 25, 16, 67, 68, 69, 0, 71, 72, 73, 24

Offset: 0

Henry Bottomley, Nov 25 2004

Also n^(20k+1) mod 100 for any positive integer k.
There are 63 numbers (A075821) where the final two digits of n^21, n^41, n^61, etc. are equal to n.
Period 100.

			a(11) = 11 since 11^21 = 7400249944258160101211 and the final two digits are 11.

Jianing Song, Table of n, a(n) for n = 0..99 (a period)

Cf. A010809, A051126, A075821.

[n^21 mod 100: n in [0..1000]]; // Vincenzo Librandi, Apr 21 2011

Table[Mod[n^21,100],{n,0,100}] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2011 *)
PowerMod[Range[0,80],21,100] (* Harvey P. Dale, Mar 15 2015 *)

a(n)=n^21%100 \\ Charles R Greathouse IV, Apr 06 2016

a(n) = A051126(A010809(n), 100) = a(n-100).

cp's OEIS Frontend

A242038 Integer part of square root of A010809(n) = n^21.

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A244003 A(n,k) = k^Fibonacci(n); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A010810 22nd powers: a(n) = n^22.

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A075821 List of possible last two digits (leading zeros omitted) of perfect powers.

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A036099 Centered cube numbers: (n+1)^21 + n^21.

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A022537 Nexus numbers (n+1)^21 - n^21.

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A100990 a(n) = n^21 mod 100.

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