A002283 -id:A002283 - cp's OEIS frontend

A186677 Total number of positive integers below 10^n requiring 16 positive biquadrates in their representation as sum of biquadrates.

0, 4, 47, 288, 587, 874, 1178, 1487, 1803, 2089, 2388

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) + a(n) + A186680(n) + A186682(n) + A186684(n) = A002283(n).

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

Cf. A046047, A186678.

a(5)-a(6) from Lars Blomberg, May 08 2011
a(7) from Charles R Greathouse IV, May 08 2011
a(8)-a(9) from Hiroaki Yamanouchi, Oct 13 2014
a(10)-a(11) from Giovanni Resta, Apr 29 2016

A186680 Total number of positive integers below 10^n requiring 17 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

0, 3, 33, 63, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65

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) + a(n) + A186682(n) + A186684(n) = A002283(n)
a(n) = 65 for n >= 5. - Nathaniel Johnston, May 09 2011
Continued fraction expansion of (826055+sqrt(4229))/2503382. - Bruno Berselli, May 10 2011

Eric Weisstein's World of Mathematics, Waring's Problem.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A046048, A099591.

PadRight[{0, 3, 33, 63}, 100, 65] (* Paolo Xausa, Jul 31 2024 *)

G.f.: x^2*(3+30*x+30*x^2+2*x^3)/(1-x). - Bruno Berselli, May 10 2011

a(5)-a(6) from Lars Blomberg, May 08 2011
a(7) from Charles R Greathouse IV, May 08 2011
Terms after a(7) from Nathaniel Johnston, May 09 2011

A186682 Total number of positive integers below 10^n requiring 18 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

0, 2, 19, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24

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) + a(n) + A186684(n) = A002283(n)
a(n) = 24 for n >= 4. - Nathaniel Johnston, May 09 2011
Continued fraction expansion of (185-sqrt(145))/355. - Bruno Berselli, May 10 2011

Eric Weisstein's World of Mathematics, Waring's Problem.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A046049.

PadRight[{0, 2, 19}, 100, 24] (* Paolo Xausa, Jul 30 2024 *)

G.f.: x^2*(2+17*x+5*x^2)/(1-x). - Bruno Berselli, May 10 2011

a(5)-a(6) from Lars Blomberg, May 08 2011
a(7) from Charles R Greathouse IV, May 08 2011
Terms after a(7) from Nathaniel Johnston, May 09 2011

A102347 Number of distinct prime factors of 10^n - 1.

Original entry on oeis.org

1, 2, 2, 3, 3, 5, 3, 5, 3, 5, 3, 7, 4, 5, 6, 7, 3, 8, 2, 8, 7, 7, 2, 10, 6, 7, 5, 9, 6, 13, 4, 12, 6, 7, 8, 11, 4, 4, 6, 12, 5, 14, 5, 11, 9, 7, 3, 13, 5, 11, 8, 10, 5, 12, 9, 13, 6, 9, 3, 20, 8, 6, 13, 16, 8, 14, 4, 11, 6, 13, 3, 17, 4, 8, 12, 7, 9, 15, 7, 16, 10, 8, 4, 21, 8, 9, 10, 15, 6, 21, 13

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Feb 20 2005

nonn

Table of n, a(n) for n = 1..352

Cf. A001221, A002283, A070528.

A102347 := proc(n)
    10^n-1 ;
    A001221(%) ;
end proc: # R. J. Mathar, Dec 02 2016

Table[PrimeNu[10^n-1],{n,100}] (* The program will take a long time to execute *) (* Harvey P. Dale, Jan 18 2015 *)

a(n) = omega(10^n-1); \\ Michel Marcus, Apr 22 2017

a(n) = A001221(A002283(n)) = A001221(10^n - 1).

a(n) = A001221(R_n) + (n^2 mod 3) = A095370(n) + (n^2 mod 3), where R_n = (10^n-1)/9 = A002275(n). That is, a(n) = A095370(n) for n=3k; otherwise a(n) = A095370(n) + 1. - Lekraj Beedassy, Jun 09 2006

Terms to a(280) and a(323)-a(352) in b-file from Max Alekseyev, Dec 28 2011, Apr 26 2022

a(281)-a(322) in b-file from Ray Chandler, Apr 22 2017

A070528 Number of divisors of 10^n-1 (999...999 with n digits).

Original entry on oeis.org

3, 6, 8, 12, 12, 64, 12, 48, 20, 48, 12, 256, 24, 48, 128, 192, 12, 640, 6, 384, 256, 288, 6, 2048, 96, 192, 96, 768, 96, 16384, 24, 6144, 128, 192, 384, 5120, 24, 24, 128, 6144, 48, 49152, 48, 4608, 1280, 192, 12, 16384, 48, 3072, 512, 1536, 48, 12288, 768

Offset: 1

Henry Bottomley, May 02 2002

			a(7)=12 since the divisors of 9999999 are 1, 3, 9, 239, 717, 2151, 4649, 13947, 41841, 1111111, 3333333, 9999999.

Table of n, a(n) for n = 1..352

Cf. A004023 (indices of 6's).
Cf. A018282, A018766, A027894, A027893, A027892, A027891, A027890, A027889, A027895.
Cf. A046801, A005422, A102347, A046053, A057951, A007138, A003060, A059892, A085035.

DivisorSigma[0,#]&/@(10^Range[60]-1) (* Harvey P. Dale, Jan 14 2011 *)
Table[DivisorSigma[0, 10^n - 1], {n, 60}] (* T. D. Noe, Aug 18 2011 *)

a(n) = numdiv(10^n - 1); \\ Michel Marcus, Sep 08 2015

a(n) = A000005(A002283(n)).
a(n) = Sum_{d|n} A059892(d).
a(n) = A070529(n)*(A007949(n)+3)/(A007949(n)+1).

Terms to a(280) in b-file from Hans Havermann, Aug 19 2011
a(281)-a(322) in b-file from Ray Chandler, Apr 22 2017
a(323)-a(352) in b-file from Max Alekseyev, May 04 2022

A332190 a(n) = 10^(2n+1) - 1 - 9*10^n.

Original entry on oeis.org

0, 909, 99099, 9990999, 999909999, 99999099999, 9999990999999, 999999909999999, 99999999099999999, 9999999990999999999, 999999999909999999999, 99999999999099999999999, 9999999999990999999999999, 999999999999909999999999999, 99999999999999099999999999999, 9999999999999990999999999999999

Offset: 0

M. F. Hasler, Feb 08 2020

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002283 (9*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A332120 .. A332180 (variants with different repeated digit 2, ..., 8).
Cf. A332191 .. A332197, A181965 (variants with different middle digit 1, ..., 8).

Maple
```
A332190 := n -> 10^(2*n+1)-1-9*10^n;
```

Array[10^(2 # + 1)-1-9*10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{0,909,99099},20] (* Harvey P. Dale, May 28 2021 *)

apply( {A332190(n)=10^(n*2+1)-1-9*10^n}, [0..15])

def A332190(n): return 10**(n*2+1)-1-9*10^n

a(n) = 9*A138148(n) = A002283(2n+1) - A011557(n).
G.f.: 9*x*(101 - 200*x)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A332197 a(n) = 10^(2n+1) - 1 - 2*10^n.

Original entry on oeis.org

7, 979, 99799, 9997999, 999979999, 99999799999, 9999997999999, 999999979999999, 99999999799999999, 9999999997999999999, 999999999979999999999, 99999999999799999999999, 9999999999997999999999999, 999999999999979999999999999, 99999999999999799999999999999, 9999999999999997999999999999999

Offset: 0

M. F. Hasler, Feb 08 2020

According to Kamada, n = 118 and n = 145126 are the only known indices of primes (the so-called palindromic near-repdigit or wing primes).

Patrick De Geest, Palindromic Wing Primes: (9)7(9), updated June 25, 2017.
Makoto Kamada, Factorization of 99...99799...99, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002283 (9*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only).
Cf. A332190 .. A332196, A181965 (variants with different middle digit 0, ..., 8).
Cf. A332117 .. A332187 (variants with different repeated digit 1, ..., 9).

Maple
```
A332197 := n -> 10^(n*2+1)-1-2*10^n;
```

Array[ 10^(2 # + 1) -1 -2*10^# &, 15, 0]
Table[FromDigits[Join[PadRight[{},n,9],{7},PadRight[{},n,9]]],{n,0,20}] (* or *) LinearRecurrence[{111,-1110,1000},{7,979,99799},20] (* Harvey P. Dale, Mar 03 2023 *)

apply( {A332197(n)=10^(n*2+1)-1-2*10^n}, [0..15])

def A332197(n): return 10**(n*2+1)-1-2*10^n

a(n) = 9*A138148(n) + 7*10^n.
G.f.: (7 + 202*x - 1100*x^2)/((1 - x)*(1 - 10*x)*(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A061439 Largest number whose cube has n digits.

Original entry on oeis.org

2, 4, 9, 21, 46, 99, 215, 464, 999, 2154, 4641, 9999, 21544, 46415, 99999, 215443, 464158, 999999, 2154434, 4641588, 9999999, 21544346, 46415888, 99999999, 215443469, 464158883, 999999999, 2154434690, 4641588833, 9999999999

Offset: 1

Amarnath Murthy, May 03 2001

a(n) + A181375(n) + A181377(n) + A181379(n) + A181381(n) + A181400(n) + A181402(n) + A181404(n) + A130130(n) = A002283(n).

			a(5) = 46 because 46^3 = 97336 has 5 digits, while 47^3 = 103823 has 6 digits.

a(n) is one more than the corresponding term of A018005. Cf. A061435.

Digits := 100:
A061439 := n->ceil(10^(n/3))-1:
seq (A061439(n), n=1..40);

t={}; i=0; Do[i=i+1; While[IntegerLength[i^3]<=n,i++]; AppendTo[t,i-1],{n,20}]; t (* Jayanta Basu, May 19 2013 *)

a(n) = ceiling(10^(n/3)) - 1. - Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 30 2003

More terms from Larry Reeves (larryr(AT)acm.org), May 16 2001

Typo in Maple program fixed by Martin Renner, Jan 31 2011

A178630 a(n) = 18*((10^n - 1)/9)^2.

Original entry on oeis.org

18, 2178, 221778, 22217778, 2222177778, 222221777778, 22222217777778, 2222222177777778, 222222221777777778, 22222222217777777778, 2222222222177777777778, 222222222221777777777778, 22222222222217777777777778, 2222222222222177777777777778, 222222222222221777777777777778

Offset: 1

Reinhard Zumkeller, May 31 2010

			n=1: ..................... 18 = 9 * 2;
n=2: ................... 2178 = 99 * 22;
n=3: ................. 221778 = 999 * 222;
n=4: ............... 22217778 = 9999 * 2222;
n=5: ............. 2222177778 = 99999 * 22222;
n=6: ........... 222221777778 = 999999 * 222222;
n=7: ......... 22222217777778 = 9999999 * 2222222;
n=8: ....... 2222222177777778 = 99999999 * 22222222;
n=9: ..... 222222221777777778 = 999999999 * 222222222.

G. C. Greubel, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A059988, A075412, A075415, A178631, A178632, A178633, A178634, A178635.

Cf. A002276, A002281, A002283, A002477.

List([1..20], n -> 18*((10^n-1)/9)^2); # G. C. Greubel, Jan 28 2019

[18*((10^n - 1)/9)^2: n in [1..20]]; // Vincenzo Librandi, Dec 28 2010

Table[18*((10^n-1)/9)^2, {n, 1, 20}] (* G. C. Greubel, Jan 28 2019 *)

a(n)=18*(10^n\9)^2 \\ Charles R Greathouse IV, Aug 21 2011

[18*((10^n-1)/9)^2 for n in (1..20)] # G. C. Greubel, Jan 28 2019

a(n) = 18*A002477(n) = A002283(n)*A002276(n).
a(n)=((A002276(n-1)*10 + 1)*10^(n-1) + A002281(n-1))*10 + 8.
G.f.: 18*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). - Ilya Gutkovskiy, Feb 24 2017
From Elmo R. Oliveira, Jul 30 2025: (Start)
E.g.f.: 2*exp(x)*(1 - 2*exp(9*x) + exp(99*x))/9.
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 3. (End)

A181965 a(n) = 10^(2n+1) - 10^n - 1.

Original entry on oeis.org

8, 989, 99899, 9998999, 999989999, 99999899999, 9999998999999, 999999989999999, 99999999899999999, 9999999998999999999, 999999999989999999999, 99999999999899999999999, 9999999999998999999999999, 999999999999989999999999999, 99999999999999899999999999999, 9999999999999998999999999999999

Offset: 0

Ivan Panchenko, Apr 04 2012

n 9's followed by an 8 followed by n 9's.

See A183187 = {26, 378, 1246, 1798, 2917, ...} for the indices of primes.

Patrick De Geest, Palindromic Wing Primes: (9)8(9), updated: June 25, 2017.
Makoto Kamada, Factorization of 99...99899...99, updated Dec 11 2018.
Markus Tervooren, Factorizations of (9)w8(9)w, FactorDB.com
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077794-1)/2 = A183187 (indices of primes).
Cf. A002275 (repunits R_n = (10^n-1)/9), A002283 (9*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A332190 .. A332197 (variants with different middle digit 0, ..., 7).

A181965 := n -> 10^(2*n+1)-1-10^n; # M. F. Hasler, Feb 08 2020

Array[10^(2 # + 1) - 1- 10^# &, 15, 0] (*  M. F. Hasler, Feb 08 2020 *)
Table[With[{c=PadRight[{},n,9]},FromDigits[Join[c,{8},c]]],{n,0,20}] (* Harvey P. Dale, Jun 07 2021 *)

apply( {A181965(n)=10^(n*2+1)-1-10^n}, [0..15]) \\ M. F. Hasler, Feb 08 2020

def A181965(n): return 10**(n*2+1)-1-10^n # M. F. Hasler, Feb 08 2020

From M. F. Hasler, Feb 08 2020: (Start)
a(n) = 9*A138148(n) + 8*10^n = A002283(2n+1) - A011557(10^n).
G.f.: (8 + 101*x - 1000*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2. (End)

Edited and extended to a(0) = 8 by M. F. Hasler, Feb 10 2020

cp's OEIS Frontend

A186677 Total number of positive integers below 10^n requiring 16 positive biquadrates in their representation as sum of biquadrates.

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A186680 Total number of positive integers below 10^n requiring 17 positive biquadrates in their representation as sum of biquadrates.

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A186682 Total number of positive integers below 10^n requiring 18 positive biquadrates in their representation as sum of biquadrates.

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A102347 Number of distinct prime factors of 10^n - 1.

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A070528 Number of divisors of 10^n-1 (999...999 with n digits).

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A332190 a(n) = 10^(2n+1) - 1 - 9*10^n.

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A332197 a(n) = 10^(2n+1) - 1 - 2*10^n.

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