cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

M. F. Hasler, Jan 01 2021

Keywords

Comments

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

Examples

			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...

Links

Crossrefs

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).

Programs

  • Mathematica
    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 *)
  • PARI
    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

Formula

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.

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

Original entry on oeis.org

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

Views

Author

M. F. Hasler, Jan 01 2021

Keywords

Comments

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

Examples

			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...

Crossrefs

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).

Programs

  • PARI
    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

Formula

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;

A340220 Constant whose decimal expansion is the concatenation of the largest n-digit prime A003618(n), for n = 1, 2, 3, ...

Original entry on oeis.org

7, 9, 7, 9, 9, 7, 9, 9, 7, 3, 9, 9, 9, 9, 1, 9, 9, 9, 9, 8, 3, 9, 9, 9, 9, 9, 9, 1, 9, 9, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 9, 3, 7, 9, 9, 9, 9, 9, 9, 9, 9, 6, 7, 9, 9, 9, 9, 9, 9, 9, 9, 9, 7, 7, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 7, 1
Offset: 0

Views

Author

M. F. Hasler, Jan 01 2021

Keywords

Comments

This is the limit of the terms of A338968, either digit-wise, or as a constant, up to powers of 10.

Examples

			The smallest prime with 1, 2, 3, 4, ... digits is, respectively, 7, 97, 997, 9973, 99991, 999983, ...
Here we list the sequence of digits of these numbers: 7; 9, 7; 9, 9, 7; 9, 9, 7, 3; ...
This can be considered, as for the Champernowne and Copeland-Erdős constants, as the decimal expansion of a real constant 0.797997997399991...

Crossrefs

Cf. A003618 (largest n-digit prime), A340222 (same with semiprimes), A340207 (same for squares, limit of A339978), A340209 (same for cubes, limit of A340115), A340219 (similar for smallest n-digit primes, limit of A215641), A340221 (similar, with smallest semiprime, limit of A215647), A340206 (similar, with smallest n-digit squares, limit of A215689), A340208 (similar, with smallest n-digit cubes, limit of A215692), A340220 (same for primes, limit of A338968).
Cf. A033307 (Champernowne constant), A030190 (binary), A001191 (concatenation of all squares), A134724 (cubes), A033308 (primes: Copeland-Erdős constant).

Programs

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

Formula

c = 0.797997997399991999983999999199999989999999937999999996799999999977...
= Sum_{k >= 1} 10^(-k(k+1)/2)*A003618(k)
a(-n(n+1)/2) = 9 for all n >= 0, followed by increasingly more 9s.
Showing 1-3 of 3 results.

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