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 1-10 of 19 results. Next

A186669 Total number of positive integers below 10^n requiring 12 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

0, 6, 68, 724, 7126, 67537, 667927, 6668357, 66668807, 666669258, 6666669682
Offset: 1

Views

Author

Martin Renner, Feb 25 2011

Keywords

Comments

A114322(n) + A186649(n) + A186651(n) + A186653(n) + A186655(n) + A186657(n) + A186659(n) + A186661(n) + A186663(n) + A186665(n) + A186667(n) + a(n) + A186671(n) + A186673(n) + A186675(n) + A186677(n) + A186680(n) + A186682(n) + A186684(n) = A002283(n).

Links

Crossrefs

Cf. A003346, A186670.

Extensions

a(5)-a(7) from Lars Blomberg, May 08 2011
a(8)-a(9) from Hiroaki Yamanouchi, Oct 13 2014
a(10)-a(11) from Giovanni Resta, Apr 29 2016

A186684 Total number of positive integers below 10^n requiring 19 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

0, 1, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7
Offset: 1

Views

Author

Martin Renner, Feb 25 2011

Keywords

Comments

A114322(n) + A186649(n) + A186651(n) + A186653(n) + A186655(n) + A186657(n) + A186659(n) + A186661(n) + A186663(n) + A186665(n) + A186667(n) + A186669(n) + A186671(n) + A186673(n) + A186675(n) + A186677(n) + A186680(n) + A186682(n) + a(n) = A002283(n).

References

  • J.-M. Deshouillers, K. Kawada, and T. D. Wooley, On sums of sixteen biquadrates, Mem. Soc. Math. Fr. 100 (2005), p. 120.

Links

Crossrefs

Cf. A010727, A046050.

Programs

  • Mathematica
    PadRight[{0, 1}, 100, 7] (* Paolo Xausa, Jul 30 2024 *)

Formula

a(n) = 7 for n >= 3. - Nathaniel Johnston, May 09 2011
From Elmo R. Oliveira, Aug 05 2024: (Start)
G.f.: x^2*(1 + 6*x)/(1 - x).
E.g.f.: 7*(exp(x) - 1 - x) - 3*x^2. (End)

Extensions

a(5)-a(6) from Lars Blomberg, May 08 2011
Terms after a(6) from Nathaniel Johnston, May 09 2011

A114322 Largest number whose 4th power has n digits.

Original entry on oeis.org

1, 3, 5, 9, 17, 31, 56, 99, 177, 316, 562, 999, 1778, 3162, 5623, 9999, 17782, 31622, 56234, 99999, 177827, 316227, 562341, 999999, 1778279, 3162277, 5623413, 9999999, 17782794, 31622776, 56234132, 99999999, 177827941, 316227766, 562341325, 999999999, 1778279410
Offset: 1

Views

Author

Jonathan Vos Post, Feb 06 2006

Keywords

Comments

This is to 4th powers as A061439 is to cubes and A049416 is to squares.
a(n) + A186649(n) + A186651(n) + A186653(n) + A186655(n) + A186657(n) + A186659(n) + A186661(n) + A186663(n) + A186665(n) + A186667(n) + A186669(n) + A186671(n) + A186673(n) + A186675(n) + A186677(n) + A186680(n) + A186682(n) + A186684(n) = A002283(n).

Examples

			a(10) = 316 because 316^4 = 9971220736 which has 10 digits, while 317^4 = 10098039121 has 11 digits.
a(35) = 562341325 because 562341325^4 = 99999999864602459914272843469140625 has 35 digits, while 562341326^4 = 100000000575914225104884587789852176 has 36.

Links

Crossrefs

Cf. A061439, A049416.

Programs

  • Magma
    [Ceiling((10^n)^(1/4))-1: n in [1..40]]; // Vincenzo Librandi, Oct 01 2011
  • Mathematica
    Ceiling[(10^Range[50])^(1/4)] - 1 (* Paolo Xausa, Jul 30 2024 *)

Formula

a(n) = ceiling((10^n)^(1/4)) - 1.

A186649 Total number of positive integers below 10^n requiring 2 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

1, 5, 14, 43, 143, 460, 1467, 4613, 14629, 46341, 146545, 463344, 1465658, 4634967, 14657277, 46350371
Offset: 1

Views

Author

Martin Renner, Feb 25 2011

Keywords

Comments

A114322(n) + a(n) + A186651(n) + A186653(n) + A186655(n) + A186657(n) + A186659(n) + A186661(n) + A186663(n) + A186665(n) + A186667(n) + A186669(n) + A186671(n) + A186673(n) + A186675(n) + A186677(n) + A186680(n) + A186682(n) + A186684(n) = A002283(n).

Links

Crossrefs

Cf. A003336.

Programs

  • Maple
    isbiquadrate:=proc(n) type(root(n,4),posint); end:
isA003336:=proc(n) local x,y4; if isbiquadrate(n) then false; else for x from 1 do y4:=n-x^4; if y4A003336(k) then i:=i+1; fi; od: return(i); end: for n from 1 do print(a(n)); od;

Extensions

a(6) from Martin Renner, Feb 26 2011
a(7)-a(16) from Lars Blomberg, May 08 2011

A186651 Total number of positive integers below 10^n requiring 3 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

1, 6, 29, 141, 789, 4353, 24173, 134679, 756947, 4258210, 23958106
Offset: 1

Views

Author

Martin Renner, Feb 25 2011

Keywords

Comments

A114322(n) + A186649(n) + a(n) + A186653(n) + A186655(n) + A186657(n) + A186659(n) + A186661(n) + A186663(n) + A186665(n) + A186667(n) + A186669(n) + A186671(n) + A186673(n) + A186675(n) + A186677(n) + A186680(n) + A186682(n) + A186684(n) = A002283(n).

Links

Crossrefs

Cf. A003337.
Cf. A002283, A114322, A186649, A186653, A186655, A186657, A186659, A186661, A186663, A186665, A186667, A186669, A186671, A186673, A186675, A186677, A186680, A186682, A186684.

Extensions

a(5)-a(11) from Lars Blomberg, May 08 2011

A186653 Total number of positive integers below 10^n requiring 4 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

1, 7, 48, 346, 3066, 27754, 260724, 2516312, 24744689, 245221669, 2442288495
Offset: 1

Views

Author

Martin Renner, Feb 25 2011

Keywords

Comments

A114322(n) + A186649(n) + A186651(n) + a(n) + A186655(n) + A186657(n) + A186659(n) + A186661(n) + A186663(n) + A186665(n) + A186667(n) + A186669(n) + A186671(n) + A186673(n) + A186675(n) + A186677(n) + A186680(n) + A186682(n) + A186684(n) = A002283(n)

Links

Crossrefs

Cf. A003338, A186654.

Extensions

a(5)-a(9) from Lars Blomberg, May 08 2011
a(10)-a(11) from Giovanni Resta, Apr 26 2016

A186655 Total number of positive integers below 10^n requiring 5 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

1, 7, 67, 654, 7931, 99401, 1251135, 15368035, 181588299, 2070553183, 22980757817
Offset: 1

Views

Author

Martin Renner, Feb 25 2011

Keywords

Comments

A114322(n) + A186649(n) + A186651(n) + A186653(n) + a(n) + A186657(n) + A186659(n) + A186661(n) + A186663(n) + A186665(n) + A186667(n) + A186669(n) + A186671(n) + A186673(n) + A186675(n) + A186677(n) + A186680(n) + A186682(n) + A186684(n) = A002283(n)

Links

Crossrefs

Cf. A003339, A186656.

Extensions

a(5)-a(9) from Lars Blomberg, May 08 2011
a(10)-a(11) from Giovanni Resta, Apr 26 2016

A186657 Total number of positive integers below 10^n requiring 6 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

1, 7, 78, 955, 12392, 152571, 1720133, 18164402, 177157454, 1625380302, 14433344661
Offset: 1

Views

Author

Martin Renner, Feb 25 2011

Keywords

Comments

A114322(n) + A186649(n) + A186651(n) + A186653(n) + A186655(n) + a(n) + A186659(n) + A186661(n) + A186663(n) + A186665(n) + A186667(n) + A186669(n) + A186671(n) + A186673(n) + A186675(n) + A186677(n) + A186680(n) + A186682(n) + A186684(n) = A002283(n)

Links

Crossrefs

Cf. A003340, A186658.

Extensions

a(5)-a(9) from Lars Blomberg, May 08 2011
a(10)-a(11) from Giovanni Resta, Apr 26 2016

A186659 Total number of positive integers below 10^n requiring 7 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

1, 6, 83, 1101, 12813, 135273, 1242976, 10072614, 81753935, 720313380, 6785033130
Offset: 1

Views

Author

Martin Renner, Feb 25 2011

Keywords

Comments

A114322(n) + A186649(n) + A186651(n) + A186653(n) + A186655(n) + A186657(n) + a(n) + A186661(n) + A186663(n) + A186665(n) + A186667(n) + A186669(n) + A186671(n) + A186673(n) + A186675(n) + A186677(n) + A186680(n) + A186682(n) + A186684(n) = A002283(n).

Links

Crossrefs

Cf. A003341, A186660.

Extensions

a(5)-a(8) from Lars Blomberg, May 08 2011
a(9) from Hiroaki Yamanouchi, Oct 13 2014
a(10)-a(11) from Giovanni Resta, Apr 26 2016

A186663 Total number of positive integers below 10^n requiring 9 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

1, 6, 78, 920, 8675, 77225, 691255, 6705498, 66719462, 666733325, 6666747237
Offset: 1

Views

Author

Martin Renner, Feb 25 2011

Keywords

Comments

A114322(n) + A186649(n) + A186651(n) + A186653(n) + A186655(n) + A186657(n) + A186659(n) + A186661(n) + a(n) + A186665(n) + A186667(n) + A186669(n) + A186671(n) + A186673(n) + A186675(n) + A186677(n) + A186680(n) + A186682(n) + A186684(n) = A002283(n)

Links

Crossrefs

Cf. A003343, A186664.

Extensions

a(5)-a(8) from Lars Blomberg, May 08 2011
a(9) from Hiroaki Yamanouchi, Oct 13 2014
a(10)-a(11) from Giovanni Resta, Apr 26 2016
Showing 1-10 of 19 results. Next

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