A061435 -id:A061435 - cp's OEIS frontend

A340209 Constant whose decimal expansion is the concatenation of the largest n-digit cube A061435(n), for n = 1, 2, 3, ...

8, 6, 4, 7, 2, 9, 9, 2, 6, 1, 9, 7, 3, 3, 6, 9, 7, 0, 2, 9, 9, 9, 9, 3, 8, 3, 7, 5, 9, 9, 8, 9, 7, 3, 4, 4, 9, 9, 7, 0, 0, 2, 9, 9, 9, 9, 9, 9, 3, 9, 4, 8, 2, 6, 4, 9, 9, 9, 6, 1, 9, 4, 6, 7, 2, 1, 9, 9, 9, 7, 0, 0, 0, 2, 9, 9, 9, 9, 9, 9, 9, 9, 5

Offset: 0

M. F. Hasler, Jan 01 2021

The terms of sequence A340115 converge to this sequence of digits, and to this constant, up to powers of 10.

			The largest cube with 1, 2, 3, 4, ... digits is, respectively, 8 = 2^3, 64 = 4^3, 729 = 9^3, 9261 = 21^3, ..., cf. A061435.
Here we list the sequence of digits of these numbers: 8; 6, 4; 7, 2, 9; 9, 2, 6, 1; ...
This can be considered, as for the Champernowne and Copeland-Erdős constants, as the decimal expansion of a real constant 0.864729926...

Cf. A061435 (largest n-digit cube), A340115 (has this as "limit"), A340208 (similar, with smallest n-digit cubes, limit of A215692), A340207 (same for squares, limit of A339978), A340220 (same for primes), A340222 (same for semiprimes), A340219 (similar, with smallest primes, limit of A215641), A340221 (similar, with smallest semiprimes, limit of A215647).

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

concat([digits(sqrtnint(10^k-1,3)^3)|k<-[1..14]]) \\ as seq. of digits
c(N=20)=sum(k=1,N,.1^(k*(k+1)/2)*sqrtnint(10^k-1,3)^3) \\ as constant

c = 0.86472992619733697029999383759989734499700299999939482649996194672...
  = Sum_{k >= 1} 10^(-k(k+1)/2)*floor(10^(k/3)-1)^3
a(-n(n+1)/2) = 9 for all n >= 3;

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

A340115 Largest prime whose decimal expansion consists of the concatenation of a 1-digit cube, a 2-digit cube, a 3-digit cube, ..., and an n-digit cube, or 0 if there is no such prime.

Original entry on oeis.org

0, 827, 164729, 8642164913, 864729685979507, 864729926197336531441, 8647299261973369702994826809, 864729926197336970299980034443986977, 864729926197336970299993837599897344909853209, 8647299261973369702999938375998973449970029998036054027

Offset: 1

Bernard Schott, Dec 28 2020

If a(n) exists it has A000217(n) = n*(n+1)/2 digits.
The similar smallest primes are in A215692.
We can conjecture that a(n) > 0 for all n > 1 and the terms converge to the concatenation of (c(1), c(2), c(3), ...) where c(k) is the largest k digit cube. The number of such primes between A215692(n) and a(n) is (0, 2, 2, 9, 177, 6909, 570166, ...). This is very close to what we expect given the number of concatenations of cubes of the respective length (product of 10^(k/3)-10^((k-1)/3), k=1..n) and the density of primes in that range according to the PNT. - M. F. Hasler, Dec 31 2020

			a(1) = 0 because no 1-digit cube {0, 1, 8} is prime.
a(2) = 827 because 827 is prime and is the concatenation of 8 = 2^3 and 27 = 3^3.
a(3) = 164729 because 827343, 827729, 864343 and 864729 are not primes and 164729, concatenation of 1 = 1^3, 64 = 4^3 and 729 = 9^3 is prime.

M. F. Hasler, Table of n, a(n) for n = 1..44 (all terms < 10^1000), Dec 31 2020

Cf. A000217, A000578, A003618, A061435.

Cf. A338968 (with concatenated primes), A339978 (with concatenated squares).

A340115(n)=forvec(v=vector(n,k,-[sqrtnint(10^k-1,3),ceil(10^((k-1)/3))]),ispseudoprime(n=eval(concat([Str(-k^3)|k<-v])))&&return(n)) \\ M. F. Hasler, Dec 31 2020

from sympy import isprime
from itertools import product
def a(n):
  cubes = [str(k**3) for k in range(1, int((10**n)**(1/3))+2)]
  revcbs = [[k3 for k3 in cubes if len(k3)==i+1][::-1] for i in range(n)]
  for t in product(*revcbs):
    intt = int("".join(t))
    if isprime(intt): return intt
  return 0
print([a(n) for n in range(1, 11)]) # Michael S. Branicky, Dec 28 2020

a(4)-a(10) from Michael S. Branicky, Dec 28 2020

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

A340209 Constant whose decimal expansion is the concatenation of the largest n-digit cube A061435(n), for n = 1, 2, 3, ...

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A061439 Largest number whose cube has n digits.

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A340115 Largest prime whose decimal expansion consists of the concatenation of a 1-digit cube, a 2-digit cube, a 3-digit cube, ..., and an n-digit cube, or 0 if there is no such prime.

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

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