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.

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A340206 Constant whose decimal expansion is the concatenation of the smallest n-digit square A061432(n), for n = 1, 2, 3, ...

Original entry on oeis.org

1, 1, 6, 1, 0, 0, 1, 0, 2, 4, 1, 0, 0, 0, 0, 1, 0, 0, 4, 8, 9, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 4, 5, 6, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 4, 1, 2, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 4, 7, 9, 8, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 1, 4, 9, 2, 8, 4, 1, 0, 0, 0, 0
Offset: 0

Views

Author

M. F. Hasler, Dec 31 2020

Keywords

Comments

The terms of sequence A215689 have this as limit, digit-wise and as a constant, up to powers of 10.
Every other "smallest n-digit square" (i.e., for odd n = 2k + 1) is 10^k, which explains the chunks of (1,0,...,0), cf. formula.

Examples

			The smallest square with 1, 2, 3, 4, ... digits is, respectively, 1, 16 = 4^2, 100 = 10^2, 1024 = 32^2, ....
Here we list the digits of these numbers: 1; 1, 6; 1, 0, 0; 1, 0, 2, 4; ...
As for the Champernowne and Copeland-Erdős constants, we can consider this as the decimal expansion of the real constant 0.116100102410000100489...

Crossrefs

Cf. A061432 (smallest n-digit squares), A215689 (has this as "limit"), A340207 (same for largest n-digit squares), A340208 (same for cubes), A340219 (same for primes), A340221 (same for semiprimes).
Cf. A033307 (Champernowne constant), A030190 (binary), A001191 (concatenation of all squares), A134724 (cubes), A033308 (primes: Copeland-Erdős constant).

Programs

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

Formula

c = 0.11610010241000010048910000001000456910000000010000141291000000000010...
= Sum_{k >= 1} 10^(-k(k+1)/2)*ceiling(10^((k-1)/2))^2
a(-n(n+1)/2) = 1 for all n >= 0; a(k) = 0 for -n(n-1)/2 > k > -n(n+1)/2 with odd n.

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.

A340222 Constant whose decimal expansion is the concatenation of the largest n-digit semiprime A098450(n), for n = 1, 2, 3, ...

Original entry on oeis.org

9, 9, 5, 9, 9, 8, 9, 9, 9, 8, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 7, 9, 9, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 7, 9, 9, 9, 9, 9, 9, 9, 9, 1, 9, 9, 9, 9, 9, 9, 9, 9, 9, 7, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 7, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 7, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9
Offset: 0

Views

Author

M. F. Hasler, Jan 01 2021

Keywords

Examples

			The smallest prime with 1, 2, 3, 4, ... digits is, respectively, 9 = 3^2, 95 = 5*19, 998 = 2*499, 9998 = 2*4999, .... Here we list the sequence of digits of these numbers: 9: 9, 5; 9, 9, 8; 9, 9, 9, 8; ...
This can be considered, as for the Champernowne and Copeland-Erdős constants, as the decimal expansion of a real constant 0.9959989998...

Crossrefs

Cf. A098450 (largest n-digit semiprime), A340221 (similar, with smallest semiprime, limit of A215647), A340207 (same for squares, limit of A339978), A340206 (similar, with smallest n-digit squares, limit of A215689), A340209 (same with cubes, limit of A340115), A340208 (similar, with smallest n-digit cubes, limit of A215692), A340220 (same for primes, limit of A338968), A340219 (similar for smallest n-digit primes, limit of A215641).
Cf. A033307 (Champernowne constant), A030190 (binary), A001191 (concatenation of all squares), A134724 (cubes), A033308 (primes: Copeland-Erdős constant).

Programs

  • PARI
    concat([digits(A098450(k))|k<-[1..14]]) \\ as seq. of digits
c(N=20)=sum(k=1,N,.1^(k*(k+1)/2)*A098450(k)) \\ as constant

Formula

c = 0.995998999899998999997999999899999997999999991999999999799999999997...
= Sum_{k >= 1} 10^(-k(k+1)/2)*A098450(k)
a(-n(n+1)/2) = 9 for all n >= 0, followed by increasingly more 9s.

A366033 Successive digits of consecutive terms of the prime-counting function A000720.

Original entry on oeis.org

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

Views

Author

John M. Campbell, Sep 26 2023

Keywords

Comments

By analogy with the Copeland-Erdős constant 0.2357111317... given by concatenating the base-10 expansions of consecutive entries of the sequence of prime numbers, the so-called "prime-counting Copeland-Erdős constant" 0.0122...9101011... is defined similarly, but with the use of the prime-counting function in place of the prime number sequence.

Examples

			0.012233444455666677888899999910101111...
The prime-counting function evaluated at 1 is 0, so a(0) = 0, and the first digit after the decimal point of the prime-counting Copeland-Erdős constant is 0.

Links

Crossrefs

Cf. A000720, A001191, A033307, A033308.

Programs

  • Mathematica
    Flatten[Table[IntegerDigits[PrimePi[n]], {n, 1, 57}]]
Flatten[IntegerDigits[PrimePi[Range[57]]]] (* Eric W. Weisstein, Jun 07 2024 *)
  • PARI
    concat(0, concat(vector(50, i, digits(primepi(i))))) \\ Michel Marcus, Nov 04 2023

A262549 Read A006530 digit-by-digit.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 1, 1, 3, 1, 3, 7, 5, 2, 1, 7, 3, 1, 9, 5, 7, 1, 1, 2, 3, 3, 5, 1, 3, 3, 7, 2, 9, 5, 3, 1, 2, 1, 1, 1, 7, 7, 3, 3, 7, 1, 9, 1, 3, 5, 4, 1, 7, 4, 3, 1, 1, 5, 2, 3, 4, 7, 3, 7, 5, 1, 7, 1, 3, 5, 3, 3, 1, 1, 7, 1, 9, 2, 9, 5, 9, 5, 6, 1, 3, 1, 7, 2, 1, 3, 1, 1, 6, 7, 1, 7, 2, 3, 7, 7, 1, 3, 7, 3, 3, 7, 5, 1, 9, 1, 1, 1, 3, 7, 9, 5, 3, 4, 1, 8, 3, 7, 1, 7, 4, 3
Offset: 1

Views

Author

N. J. A. Sloane, Oct 06 2015

Keywords

Links

Crossrefs

Cf. A006530, A001191, A033307.
Previous Showing 11-15 of 15 results.

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