cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A000093 a(n) = floor(n^(3/2)).

Original entry on oeis.org

0, 1, 2, 5, 8, 11, 14, 18, 22, 27, 31, 36, 41, 46, 52, 58, 64, 70, 76, 82, 89, 96, 103, 110, 117, 125, 132, 140, 148, 156, 164, 172, 181, 189, 198, 207, 216, 225, 234, 243, 252, 262, 272, 281, 291, 301, 311, 322, 332, 343, 353, 364, 374, 385, 396, 407, 419, 430
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • B. K. Agarwala and F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Integer part of square root of n^k: A000196 (k=1), this sequence (k=3), A155013 (k=5), A155014 (k=7), A155015 (k=11), A155016 (k=13), A155018 (k=15), A155019 (k=17).
Cf. A002821.
Cf. A185549.

Programs

  • Haskell
    a000093 = a000196 . a000578  -- Reinhard Zumkeller, Jul 11 2014
  • Maple
    Digits := 100: A000093 := n->floor(evalf(n^(3/2)));
  • Mathematica
    Table[ Floor[ Sqrt[n^3]], {n, 0, 60}]
  • PARI
    a(n)=if(n<0,0,sqrtint(n^3))
  • Python
    from math import isqrt
def A000093(n): return isqrt(n**3) # Chai Wah Wu, Sep 08 2024

Formula

a(n) = A077121(n) - 1. [Reinhard Zumkeller, Oct 31 2009]
a(n) = floor(n*sqrt(n)). [Arkadiusz Wesolowski, Jun 01 2011]
a(n) = A000196(A000578(n)) = A074704(n)+n*A000196(n). [Reinhard Zumkeller, Jun 27 2011]

Extensions

More terms from James Sellers, May 04 2000

A155016 Integer part of square root of A010801.

Original entry on oeis.org

0, 1, 90, 1262, 8192, 34938, 114283, 311269, 741455, 1594323, 3162277, 5875603, 10343751, 17403307, 28172943, 44115700, 67108864, 99521746, 144301645, 205068240, 286216701, 393029741, 531798888, 709955183, 936209559, 1220703125
Offset: 0

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 19 2009

Keywords

Links

Crossrefs

Integer part of square root of n^k: A000196 (k=1), A000093 (k=3), A155013 (k=5), A155014 (k=7), A155015 (k=11), this sequence (k=13), A155018 (k=15), A155019 (k=17).

Programs

  • Magma
    [Floor(Sqrt(n^13)): n in [1..30]]; // G. C. Greubel, Dec 30 2017
  • Mathematica
    a={};Do[AppendTo[a,IntegerPart[(n^13)^(1/2)]],{n,0,5!}];a
Table[Floor[Sqrt[n^13]], {n,1,30}] (* G. C. Greubel, Dec 30 2017 *)
  • PARI
    for(n=1,30, print1(floor(sqrt(n^13)), ", ")) \\ G. C. Greubel, Dec 30 2017

Extensions

Offset corrected by Karl V. Keller, Jr., Sep 27 2014

A155018 Integer part of square root of n^15 = A010803(n).

Original entry on oeis.org

0, 1, 181, 3787, 32768, 174692, 685700, 2178889, 5931641, 14348907, 31622776, 64631634, 124125023, 226242995, 394421215, 661735513, 1073741824, 1691869691, 2597429617, 3896296578, 5724334022, 8253624572, 11699575548, 16328969210
Offset: 0

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 19 2009

Keywords

Links

Crossrefs

Integer part of square root of n^k: A000196 (k=1), A000093 (k=3), A155013 (k=5), A155014 (k=7), A155015 (k=11), A155016 (k=13), this sequence (k=15), A155019 (k=17).

Programs

  • Magma
    [Floor(Sqrt(n^15)): n in [1..30]]; // G. C. Greubel, Dec 30 2017
  • Mathematica
    a={};Do[AppendTo[a,IntegerPart[(n^15)^(1/2)]],{n,0,5!}];a
Table[Floor[Sqrt[n^15]], {n,1,30}] (* G. C. Greubel, Dec 30 2017 *)
  • PARI
    for(n=1,30, print1(floor(sqrt(n^15)), ", ")) \\ G. C. Greubel, Dec 30 2017

Extensions

Offset corrected by Alois P. Heinz, Sep 27 2014

A155019 Integer part of square root of n^17 = A010805(n).

Original entry on oeis.org

0, 1, 362, 11363, 131072, 873464, 4114202, 15252229, 47453132, 129140163, 316227766, 710947978, 1489500287, 2941158941, 5521897022, 9926032708, 17179869184, 28761784747, 46753733110, 74029634996, 114486680447
Offset: 0

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 19 2009

Keywords

Links

Crossrefs

Integer part of square root of n^k: A000196 (k=1), A000093 (k=3), A155013 (k=5), A155014 (k=7), A155015 (k=11), A155016 (k=13), A155018 (k=15), this sequence (k=17).

Programs

  • Magma
    [Floor(Sqrt(n^17)): n in [0..30]]; // G. C. Greubel, Dec 30 2017
  • Mathematica
    a={};Do[AppendTo[a,IntegerPart[(n^17)^(1/2)]],{n,0,5!}];a
Table[Floor[Sqrt[n^17]], {n,0,30}] (* G. C. Greubel, Dec 30 2017 *)
  • PARI
    for(n=0,30, print1(floor(sqrt(n^17)), ", ")) \\ G. C. Greubel, Dec 30 2017

Extensions

Offset corrected by Alois P. Heinz, Sep 27 2014

A238170 Integer part of square root of A001017: a(n) = floor(n^(9/2)).

Original entry on oeis.org

0, 1, 22, 140, 512, 1397, 3174, 6352, 11585, 19683, 31622, 48558, 71831, 102978, 143739, 196069, 262144, 344365, 445375, 568056, 715541, 891223, 1098758, 1342070, 1625363, 1953125, 2330129, 2761448, 3252453, 3808824, 4436552, 5141947, 5931641, 6812597, 7792110
Offset: 0

Views

Author

Philippe Deléham, Feb 21 2014

Keywords

Links

Crossrefs

Integer part of square root of n^k: A000196 (k=1), A000093 (k=3), A155013 (k=5), A155014 (k=7), this sequence (k=9), A155015 (k=11), A155016 (k=13), A155018 (k=15), A155019 (k=17).

Programs

  • Magma
    [Floor(n^(9/2)): n in [0..40]]; // Vincenzo Librandi, Feb 23 2014
  • Mathematica
    Table[Floor[n^(9/2)], {n,0,30}] (* G. C. Greubel, Dec 30 2017 *)
  • PARI
    a(n) = floor(n^(9/2)); \\ Joerg Arndt, Feb 23 2014
  • Python
    from math import isqrt
def A238170(n): return isqrt(n**9) # Chai Wah Wu, Jan 27 2023

Formula

a(n) = floor(n^(9/2)).
a(n) = A000196(A001017(n)).
a(n) = floor(n^4*sqrt(n)).

A185542 a(n) = m*(m+1)/2, where m = floor(n^(5/2)).

Original entry on oeis.org

1, 15, 120, 528, 1540, 3916, 8385, 16471, 29646, 50086, 80601, 124251, 185745, 269011, 379756, 524800, 709836, 944625, 1237951, 1599366, 2041210, 2577585, 3216916, 3980431, 4884375, 5939181, 7172578, 8605026, 10253656, 12149985, 14313925, 16776528, 19565640, 22717170, 26263128, 30236976, 34673628
Offset: 1

Views

Author

Clark Kimberling, Jan 30 2011

Keywords

Comments

A subsequence of A000217, the triangular numbers.

Links

Crossrefs

Cf. A000217, A155013.

Programs

  • Mathematica
    f[n_] = Floor[n^(5/2)]*Floor[n^(5/2)+1]/2; Table[f[n],{n,1,80}]
(#(#+1))/2&/@Floor[Range[40]^(5/2)] (* Harvey P. Dale, Feb 27 2024 *)
  • PARI
    for(n=1,50, print1(floor(n^(5/2))*floor(1 + n^(5/2))/2, ", ")) \\ G. C. Greubel, Jul 07 2017

Formula

a(n) = A000217(A155013(n)). - Michel Marcus, Jul 08 2017

A255616 Table read by antidiagonals, T(n,k) = floor(sqrt(k^n)), n >= 0, k >=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 4, 5, 4, 1, 1, 2, 5, 8, 9, 5, 1, 1, 2, 6, 11, 16, 15, 8, 1, 1, 2, 7, 14, 25, 32, 27, 11, 1, 1, 3, 8, 18, 36, 55, 64, 46, 16, 1, 1, 3, 9, 22, 49, 88, 125, 128, 81, 22, 1, 1, 3, 10, 27, 64, 129, 216, 279, 256, 140, 32, 1, 1, 3, 11, 31, 81, 181, 343, 529, 625, 512, 243, 45, 1
Offset: 0

Views

Author

Kival Ngaokrajang, Feb 28 2015

Keywords

Examples

			See table in the links.

Links

Crossrefs

Rows(1..10): A000012, A000196, A000027, A000093, A000290, A155013, A000578, A155014, A000583, A238170.
Columns(1..10): A000012, A017910, A017913, A000079, A017919, A017922, A017925, A017928, A000244, A017934.
Diagonals: A190343, A066642, A075264.

Programs

  • Mathematica
    T[n_, k_] := Floor[Sqrt[k^n]]; Table[T[k, n + 1 - k], {n, 0, 15}, {k, 0, n}] (* G. C. Greubel, Dec 30 2017 *)
  • PARI
    {for(i=1,20,for(n=0,i-1,a=floor(sqrt((i-n)^n));print1(a,", ")))}

Formula

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

Extensions

Terms a(81) onward added by G. C. Greubel, Dec 30 2017
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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