A114322 -id:A114322 - cp's OEIS frontend

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

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

Offset: 1

Martin Renner, Feb 25 2011

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).

Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A003346, A186670.

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

Martin Renner, Feb 25 2011

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).

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

J.-M. Deshouillers, F. Hennecart and B. Landreau, Waring's Problem for sixteen biquadrates - numerical results, Journal de Théorie des Nombres de Bordeaux 12 (2000), pp. 411-422.
L. E. Dickson, Recent progress on Waring's theorem and its generalizations, Bull. Amer. Math. Soc. 39:10 (1933), pp. 701-727.
Eric Weisstein's World of Mathematics, Waring's Problem.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A010727, A046050.

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

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)

a(5)-a(6) from Lars Blomberg, May 08 2011

Terms after a(6) from Nathaniel Johnston, May 09 2011

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

Martin Renner, Feb 25 2011

nonn

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).

Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A003336.

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;

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

Martin Renner, Feb 25 2011

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).

Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A003337.

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

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

Martin Renner, Feb 25 2011

nonn

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)

Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A003338, A186654.

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

Martin Renner, Feb 25 2011

nonn

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)

Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A003339, A186656.

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

Martin Renner, Feb 25 2011

nonn

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)

Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A003340, A186658.

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

Martin Renner, Feb 25 2011

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).

Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A003341, A186660.

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

A186661 Total number of positive integers below 10^n requiring 8 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

1, 6, 83, 1058, 10515, 94474, 794880, 7015673, 67241571, 667465566, 6667690837

Offset: 1

Martin Renner, Feb 25 2011

A114322(n) + A186649(n) + A186651(n) + A186653(n) + A186655(n) + A186657(n) + A186659(n) + a(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).

Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A003342, A186662.

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

Martin Renner, Feb 25 2011

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)

Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A003343, A186664.

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

cp's OEIS Frontend

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

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

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

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

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

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

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

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

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A186661 Total number of positive integers below 10^n requiring 8 positive biquadrates in their representation as sum of biquadrates.

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