A052268 -id:A052268 - cp's OEIS frontend

A093851 a(n) = A002283(n-1) + floor(A052268(n)/(1+n)).

4, 39, 324, 2799, 24999, 228570, 2124999, 19999999, 189999999, 1818181817, 17499999999, 169230769229, 1642857142856, 15999999999999, 156249999999999, 1529411764705881, 14999999999999999, 147368421052631577, 1449999999999999999, 14285714285714285713

Offset: 1

Amarnath Murthy, Apr 18 2004

The first column r=1 of a triangle defined by T(n,r) = 10^(n-1) -1 + r*floor(9*10^(n-1)/(n+1)).

A row starts with a (virtual) 0th column of a rep-9-digit and fills the remainder with n+1 numbers in arithmetic progression with the largest step such that all numbers in the n-th row are n-digit numbers.

			The triangle starts in row n=1 as
4 9 # -1, -1+5, -1+2*5
39 69 99 # 9,9+30,9+2*30
324 549 774 999 # 99, 99+225, 99+2*225, 99+3*225
2799 4599 6399 8199 9999 # 999, 999+1800, 999+2*1800,..
...
The sequence contains the first column.

G. C. Greubel, Table of n, a(n) for n = 1..995

Cf. A061772, A093846, A093847, A093450, A093552.

[10^(n-1) -1 +Floor(9*10^(n-1)/(n+1)): n in [1..20]]; // G. C. Greubel, Apr 02 2019

A093851 := proc(n) 10^(n-1)-1+floor(9*10^(n-1)/(n+1)) ; end proc: seq(A093851(n),n=1..20) ; # R. J. Mathar, Oct 14 2010

Table[10^(n-1) -1 +Floor[9*10^(n-1)/(n+1)], {n, 1, 20}] (* G. C. Greubel, Apr 02 2019 *)

{a(n) = 10^(n-1) -1 +floor(9*10^(n-1)/(n+1))}; \\ G. C. Greubel, Apr 02 2019

[10^(n-1) -1 +floor(9*10^(n-1)/(n+1)) for n in (1..20)] # G. C. Greubel, Apr 02 2019

a(n) = 10^(n-1) -1 + floor(9*10^(n-1)/(n+1)). - G. C. Greubel, Apr 02 2019

More terms from R. J. Mathar, Oct 14 2010

A181354 Number of n-digit perfect cubes.

Original entry on oeis.org

2, 2, 5, 12, 25, 53, 116, 249, 535, 1155, 2487, 5358, 11545, 24871, 53584, 115444, 248715, 535841, 1154435, 2487154, 5358411, 11544347, 24871542, 53584111, 115443470, 248715414, 535841116, 1154434691, 2487154143, 5358411166

Offset: 1

Martin Renner, Jan 28 2011

a(n) is also the total number of n-digit numbers requiring 1 positive cube in their representation as sum of cubes.
a(n) + A181376(n) + A181378(n) + A181380(n) + A181384(n) + A181401(n) + A181403(n) + A181405(n) + A171386(n) = A052268(n).
Differs from A062941 only at n=1, because 0 is considered a 0-digit, not a 1-digit number here. - R. J. Mathar, Jul 09 2011

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

Cf. A000578

a:=n->ceil(10^(n/3))-ceil(10^((n-1)/3));

With[{c = Range[4650000]^3}, Length[#]&/@Table[Select[c, IntegerLength[#] == n &], {n, 20}]] (* Harvey P. Dale, Feb 01 2011 *)
Differences[Ceiling[10^(Range[0, 30]/3)]]

a(n) = A061439(n) - A061439(n-1).

More terms from T. D. Noe, Feb 01 2011

A186654 Total number of n-digit numbers requiring 4 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

1, 6, 41, 298, 2720, 24688, 232970, 2255588, 22228377, 220476980, 2197066826

Offset: 1

Martin Renner, Feb 25 2011

A102831(n) + A186650(n) + A186652(n) + a(n) + A186656(n) + A186658(n) + A186660(n) + A186662(n) + A186664(n) + A186666(n) + A186668(n) + A186670(n) + A186672(n) + A186674(n) + A186676(n) + A186678(n) + A186681(n) + A186683(n) + A186685(n) = A052268(n), for n>1.

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

Cf. A003338, A186653.

a(n) = A186653(n) - A186653(n-1).

a(5)-a(11) from Giovanni Resta, Apr 26 2016

A186656 Total number of n-digit numbers requiring 5 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

1, 6, 60, 587, 7277, 91470, 1151734, 14116900, 166220264, 1888964884, 20910204634

Offset: 1

Martin Renner, Feb 25 2011

A102831(n) + A186650(n) + A186652(n) + A186654(n) + a(n) + A186658(n) + A186660(n) + A186662(n) + A186664(n) + A186666(n) + A186668(n) + A186670(n) + A186672(n) + A186674(n) + A186676(n) + A186678(n) + A186681(n) + A186683(n) + A186685(n) = A052268(n), for n>1.

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

Cf. A003339, A186655.

a(n) = A186655(n) - A186655(n-1).

a(5)-a(11) from Giovanni Resta, Apr 29 2016

A186658 Total number of n-digit numbers requiring 6 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

1, 6, 71, 877, 11437, 140179, 1567562, 16444269, 158993052, 1448222849, 12807964358

Offset: 1

Martin Renner, Feb 25 2011

A102831(n) + A186650(n) + A186652(n) + A186654(n) + A186656(n) + a(n) + A186660(n) + A186662(n) + A186664(n) + A186666(n) + A186668(n) + A186670(n) + A186672(n) + A186674(n) + A186676(n) + A186678(n) + A186681(n) + A186683(n) + A186685(n) = A052268(n), for n>1.

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

Cf. A003340, A186657.

a(n) = A186657(n) - A186657(n-1).

a(5)-a(11) from Giovanni Resta, Apr 29 2016

A186660 Total number of n-digit numbers requiring 7 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

1, 5, 77, 1018, 11712, 122460, 1107703, 8829638, 71681321, 638559445, 6064719750

Offset: 1

Martin Renner, Feb 25 2011

A102831(n) + A186650(n) + A186652(n) + A186654(n) + A186656(n) + A186658(n) + a(n) + A186662(n) + A186664(n) + A186666(n) + A186668(n) + A186670(n) + A186672(n) + A186674(n) + A186676(n) + A186678(n) + A186681(n) + A186683(n) + A186685(n) = A052268(n), for n>1.

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

Cf. A003341, A186659.

a(n) = A186659(n) - A186659(n-1).

a(5)-a(11) from Giovanni Resta, Apr 29 2016

A186662 Total number of n-digit numbers requiring 8 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

1, 5, 77, 975, 9457, 83959, 700406, 6220793, 60225898, 600223995, 6000225272

Offset: 1

Martin Renner, Feb 25 2011

A102831(n) + A186650(n) + A186652(n) + A186654(n) + A186656(n) + A186658(n) + A186660(n) + a(n) + A186664(n) + A186666(n) + A186668(n) + A186670(n) + A186672(n) + A186674(n) + A186676(n) + A186678(n) + A186681(n) + A186683(n) + A186685(n) = A052268(n), for n>1.

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

Cf. A003342, A186661.

a(n) = A186661(n) - A186661(n-1).

a(5)-a(11) from Giovanni Resta, Apr 29 2016

A186664 Total number of n-digit numbers requiring 9 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

1, 5, 72, 842, 7755, 68550, 614030, 6014243, 60013964, 600013863, 6000013912

Offset: 1

Martin Renner, Feb 25 2011

A102831(n) + A186650(n) + A186652(n) + A186654(n) + A186656(n) + A186658(n) + A186660(n) + A186662(n) + a(n) + A186666(n) + A186668(n) + A186670(n) + A186672(n) + A186674(n) + A186676(n) + A186678(n) + A186681(n) + A186683(n) + A186685(n) = A052268(n), for n>1.

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

Cf. A003343, A186663.

a(n) = A186663(n) - A186663(n-1).

a(5)-a(11) from Giovanni Resta, Apr 29 2016

A186666 Total number of n-digit numbers requiring 10 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

0, 6, 66, 740, 7064, 62754, 602728, 6002740, 60002625, 600002765, 6000002823

Offset: 1

Martin Renner, Feb 25 2011

A102831(n) + A186650(n) + A186652(n) + A186654(n) + A186656(n) + A186658(n) + A186660(n) + A186662(n) + A186664(n) + a(n) + A186668(n) + A186670(n) + A186672(n) + A186674(n) + A186676(n) + A186678(n) + A186681(n) + A186683(n) + A186685(n) = A052268(n), for n>1.

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

Cf. A003344, A186665.

a(n) = A186665(n) - A186665(n-1).

a(5)-a(11) from Giovanni Resta, Apr 29 2016

A186668 Total number of n-digit numbers requiring 11 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

0, 6, 64, 690, 6674, 60914, 600941, 6000995, 60000942, 600000942, 6000000909

Offset: 1

Martin Renner, Feb 25 2011

A102831(n) + A186650(n) + A186652(n) + A186654(n) + A186656(n) + A186658(n) + A186660(n) + A186662(n) + A186664(n) + A186666(n) + a(n) + A186670(n) + A186672(n) + A186674(n) + A186676(n) + A186678(n) + A186681(n) + A186683(n) + A186685(n) = A052268(n), for n>1.

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

Cf. A003345, A186667.

a(n) = A186667(n) - A186667(n-1).

a(5)-a(11) from Giovanni Resta, Apr 29 2016

cp's OEIS Frontend

A093851 a(n) = A002283(n-1) + floor(A052268(n)/(1+n)).

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A181354 Number of n-digit perfect cubes.

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A186654 Total number of n-digit numbers requiring 4 positive biquadrates in their representation as sum of biquadrates.

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A186656 Total number of n-digit numbers requiring 5 positive biquadrates in their representation as sum of biquadrates.

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A186658 Total number of n-digit numbers requiring 6 positive biquadrates in their representation as sum of biquadrates.

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A186660 Total number of n-digit numbers requiring 7 positive biquadrates in their representation as sum of biquadrates.

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A186662 Total number of n-digit numbers requiring 8 positive biquadrates in their representation as sum of biquadrates.

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A186664 Total number of n-digit numbers requiring 9 positive biquadrates in their representation as sum of biquadrates.

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