A061433 -id:A061433 - cp's OEIS frontend

A340207 Constant whose decimal expansion is the concatenation of the largest n-digit square A061433(n), for n = 1, 2, 3, ...

9, 8, 1, 9, 6, 1, 9, 8, 0, 1, 9, 9, 8, 5, 6, 9, 9, 8, 0, 0, 1, 9, 9, 9, 8, 2, 4, 4, 9, 9, 9, 8, 0, 0, 0, 1, 9, 9, 9, 9, 5, 0, 8, 8, 4, 9, 9, 9, 9, 8, 0, 0, 0, 0, 1, 9, 9, 9, 9, 9, 5, 1, 5, 5, 2, 9, 9, 9, 9, 9, 9, 8, 0, 0, 0, 0, 0, 1, 9, 9, 9, 9, 9, 9, 5, 8

Offset: 0

M. F. Hasler, Jan 01 2021

The terms of sequence A339978 converge to this sequence of digits, and to this constant, up to powers of 10.

			The largest square with 1, 2, 3, 4, ... digits is, respectively, 9 = 3^2, 81 = 9^2, 961 = 31^2, 9801 = 99^2, ....
Here we list the sequence of digits of these numbers: 9; 8, 1; 9, 6, 1; 9, 8, 0, 1; 9, 9, 8, 5, 6; ...
This can be considered, as for the Champernowne and Copeland-Erdős constants, as the decimal expansion of a real constant 0.98196198...

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A061433 (largest n-digit square), A339978 (has this as "limit"), A340208 (same with "smallest n-digit cube", limit of A215692), A340209 (same for cubes, limit of A340115), A340220 (same for primes), A340219 (similar, with smallest primes, limit of A215641), A340222 (same for semiprimes), A340221 (same for smallest semiprimes, limit of A215647).

Cf. A033307 (Champernowne constant), A030190 (binary), A001191 (concatenation of all squares), A134724 (cubes), A033308 (primes: Copeland-Erdős constant).

lnds[k_]:=Module[{c=Sqrt[10^k]},If[IntegerQ[c],(c-1)^2,Floor[c]^2]]; Flatten[IntegerDigits/@(lnds/@Range[15])] (* Harvey P. Dale, Dec 16 2021 *)

concat([digits(sqrtint(10^k-1)^2)|k<-[1..14]]) \\ as seq. of digits
c(N=20)=sum(k=1,N,.1^(k*(k+1)/2)*sqrtint(10^k-1)^2) \\ as constant

c = 0.9819619801998569980019998244999800019999508849999800001999995155...
  = Sum_{k >= 1} 10^(-k(k+1)/2)*floor(10^(k/2)-1)^2
a(-n(n+1)/2) = 9 for all n >= 2.

A049416 Largest number whose square has n digits.

Original entry on oeis.org

3, 9, 31, 99, 316, 999, 3162, 9999, 31622, 99999, 316227, 999999, 3162277, 9999999, 31622776, 99999999, 316227766, 999999999, 3162277660, 9999999999, 31622776601, 99999999999, 316227766016, 999999999999, 3162277660168

Offset: 1

Ulrich Schimke (ulrschimke(AT)aol.com)

a(n) + A180416(n) + A180425(n) + A167615(n) = A002283(n).

			31^2 = 961, but 32^2 = 1024, hence a(3) = 31.
a(4) = 99: 99^2 = 9801 has 4 digits, while 100^2 = 10000 has 5 digits.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A061433, A049415. Equals A017936 - 1.

[Ceiling(Sqrt(10^n))-1: n in [1..30]]; // Vincenzo Librandi, Oct 01 2011

Ceiling[Sqrt[10^Range[40]]-1] (* Harvey P. Dale, Sep 30 2011 *)

a(n) = ceiling(sqrt(10^n)) - 1.

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

A339978 a(n) is the largest prime whose decimal expansion consists of the concatenation of a 1-digit square, a 2-digit square, a 3-digit square, ..., and an n-digit square, or 0 if there is no such prime.

Original entry on oeis.org

0, 449, 981961, 9819619801, 981961980196721, 981961980199856194481, 9819619801998569980018946081, 981961980199856998001999824499740169, 981961980199856998001999824499980001989039601, 9819619801998569980019998244999800019999508849977812321

Offset: 1

Bernard Schott, Dec 25 2020

If a(n) exists it has A000217(n)= n*(n+1)/2 digits.

All the terms end with 1 or 9.

			a(1) = 0 because no 1-digit square {0, 1, 4, 9} is prime.
a(2) = 449 because 464, 481, 916, 925, 936, 949, 964, and 981 are not primes and 449, concatenation of 4 = 2^2 with 49 = 7^2, is prime.
a(4) = 9819619801, which is a prime is the concatenation of 9 = 3^2 with 81 = 9^2, then 961 = 31^2 and 9801 = 99^2. Observation, 9, 81, 961 and 9801 are the largest squares with respectively 1, 2, 3 and 4 digits.

David A. Corneth, Table of n, a(n) for n = 1..40 (first 16 terms from Michael S. Branicky)

Cf. A000290, A003618, A061433 (largest squares), A338968 (concatenate primes).

from sympy import isprime
from itertools import product
def a(n):
  squares = [str(k*k) for k in range(1, int((10**n)**.5)+2)]
  revsqrs = [[kk for kk in squares if len(kk)==i+1][::-1] for i in range(n)]
  for t in product(*revsqrs):
    intt = int("".join(t))
    if isprime(intt): return intt
  return 0
print([a(n) for n in range(1, 11)]) # Michael S. Branicky, Dec 25 2020

a(5)-a(10) from Michael S. Branicky, Dec 25 2020

A061435 a(n) is the largest n-digit cube.

Original entry on oeis.org

8, 64, 729, 9261, 97336, 970299, 9938375, 99897344, 997002999, 9993948264, 99961946721, 999700029999, 9999516957184, 99994258523375, 999970000299999, 9999934692543307, 99999429057832312, 999997000002999999

Offset: 1

Amarnath Murthy, May 03 2001

			a(4) = 9261 = 21^3 has 4 digits while 22^3 = 10648 has 5 digits.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A061432, A061433, A061434.

Maple
```
A061435 := n->(ceil(10^(n/3))-1)^3;
```

Table[Floor[Surd[10^n-1,3]]^3,{n,20}] (* Harvey P. Dale, Apr 02 2020 *)

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

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

A061434 a(n) is the smallest n-digit cube.

Original entry on oeis.org

1, 27, 125, 1000, 10648, 103823, 1000000, 10077696, 100544625, 1000000000, 10007873875, 100026577288, 1000000000000, 10000909453625, 100000721719296, 1000000000000000, 10000073940248384, 100000075387171679, 1000000000000000000, 10000004316234262875

Offset: 1

Amarnath Murthy, May 03 2001

			a(4) = 1000 = 10^3 has 4 digits while 9^3 = 729 has 3 digits.

Cf. A000578, A061432, A061433.

Maple
```
A061434 := n->ceil(10^((n-1)/3))^3;
```

Table[Ceiling[Surd[10^n,3]]^3,{n,0,20}] (* Harvey P. Dale, Oct 09 2023 *)

a(n) = ceil(10^((n-1)/3))^3; \\ Michel Marcus, Jan 27 2021

from sympy import integer_nthroot
def a(n):
  r, exact = integer_nthroot(10**(n-1), 3)
  return 10**(n-1) if exact else (r+1)**3
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Jan 27 2021

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

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

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 30 2003

A128826 a(n) = 10^(2n-1) minus largest square less than 10^(2n-1).

Original entry on oeis.org

1, 39, 144, 1756, 49116, 484471, 4175271, 38053824, 10649244, 1064924400, 43246886799, 529955487744, 2399106211776, 50173961567511, 590207432515431, 2099745368512359, 20237877241133151, 126421128012287511

Offset: 1

Zak Seidov, Apr 12 2007

nonn

For even indices a(2k) = 2*10^k-1, hence only odd powers of 10 are considered in this sequence.

			a(1) = 1 because 10 - 3^2 = 1.
a(2) = 39 because 1000 - 31^2 = 39.
a(3) = 144 because 100000 - 316^2 = 144.

Hugo Pfoertner, Table of n, a(n) for n = 1..500

Cf. A061433, A051221.

Table[10^n-Floor[(10^n-1)^(1/2)]^2,{n,1,40,2}]

a(n)=10^(2n-1)-A061433(2n-1).

a(n) = 10^(2*n-1) - floor(sqrt(10^(2*n-1)))^2.

A069659 Largest n-digit perfect power.

Original entry on oeis.org

9, 81, 961, 9801, 99856, 998001, 9998244, 99980001, 999950884, 9999800001, 99999515529, 999998000001, 9999995824729, 99999980000001, 999999961946176, 9999999800000001, 99999999989350756, 999999998000000001

Offset: 1

Amarnath Murthy, Apr 04 2002

Are there terms which are not perfect squares? In other words, when (if ever) does this differ from A061433.

Trivially, 81 is both a square and a fourth power. Assuming my program works, there are no differences in the first 1500 terms. - Hans Havermann, Aug 06 2006

Cf. A061433.

More terms from Ryan Propper, Oct 10 2005

A119273 Absolute value of the difference between largest square and largest cube each with n decimal digits.

Original entry on oeis.org

1, 17, 232, 540, 2520, 27702, 59869, 82657, 2947885, 5851737, 37568808, 297970002, 478867545, 5721476626, 29961646177, 65107456694, 570931518444, 2997997000002, 9607464857096, 53858918990529, 299956723113202, 1253472906066265, 2171966135005184

Offset: 1

Zak Seidov, May 12 2006

a(n) = |A061433(n) - A061435(n)|, where A061433 and A061435 are the largest n-digit square and cube.

Harvey P. Dale, Table of n, a(n) for n = 1..90

Cf. A061433, A061435.

Table[Abs[Floor[Sqrt[10^n-1]]^2-Floor[Surd[10^n-1,3]]^3],{n,30}] (* Harvey P. Dale, Apr 24 2022 *)

a(n) = (ceiling(10^(n/2)) - 1)^2 - (ceiling(10^(n/3)) - 1)^3.

Corrected, extended, and definition clarified by Harvey P. Dale, Apr 24 2022

cp's OEIS Frontend

A340207 Constant whose decimal expansion is the concatenation of the largest n-digit square A061433(n), for n = 1, 2, 3, ...

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A049416 Largest number whose square has n digits.

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A339978 a(n) is the largest prime whose decimal expansion consists of the concatenation of a 1-digit square, a 2-digit square, a 3-digit square, ..., and an n-digit square, or 0 if there is no such prime.

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A061435 a(n) is the largest n-digit cube.

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A061434 a(n) is the smallest n-digit cube.

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A128826 a(n) = 10^(2n-1) minus largest square less than 10^(2n-1).

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A069659 Largest n-digit perfect power.

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