A033307 -id:A033307 - cp's OEIS frontend

A340219 Constant whose decimal expansion is the concatenation of the smallest n-digit prime A003617(n), for n = 1, 2, 3, ...

2, 1, 1, 1, 0, 1, 1, 0, 0, 9, 1, 0, 0, 0, 7, 1, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 1, 9, 1, 0, 0, 0, 0, 0, 0, 0, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 7, 1, 0, 0, 0

Offset: 0

M. F. Hasler, Jan 01 2021

The terms of sequence A215641 converge to this sequence of digits, and to this constant, up to powers of 10.

			The smallest prime with 1, 2, 3, 4, ... digits is, respectively, 2, 11, 101, 1009, .... Here we list the sequence of digits of these numbers: 2: 1, 1; 1, 0, 1; 1, 0, 0, 9; ...
This can be considered, as for the Champernowne and Copeland-Erdős constants, as the decimal expansion of a real constant 0.2111011009...

Cf. A003617 (smallest n-digit prime), A215641 (has this as "limit"), A340206 (same for squares, limit of A215689), A340207 (similar, with largest n-digit squares, limit of A339978), A340208 (same for cubes, limit of A215692), A340209 (same with largest n-digit cube, limit of A340115), A340221 (same for semiprimes, limit of A215647).

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

Flatten[Table[IntegerDigits[NextPrime[10^n]],{n,0,20}]] (* Harvey P. Dale, Mar 29 2024 *)

concat([digits(nextprime(10^k))|k<-[0..14]]) \\ as seq. of digits
c(N=20)=sum(k=1,N,.1^(k*(k+1)/2)*nextprime(10^(k-1))) \\ as constant

c = 0.21110110091000710000310000031000001910000000710000000071000000001...
  = Sum_{k >= 1} 10^(-k(k+1)/2)*nextprime(10^(k-1))
a(-n(n+1)/2) = 1 for all n >= 2, followed by increasingly more zeros.

A340221 Constant whose decimal expansion is the concatenation of the smallest n-digit semiprime A098449(n), for n = 1, 2, 3, ...

Original entry on oeis.org

4, 1, 0, 1, 0, 6, 1, 0, 0, 3, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 6, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

M. F. Hasler, Jan 01 2021

The terms of sequence A215647 converge to this sequence of digits, and to this constant, up to powers of 10.

			The smallest prime with 1, 2, 3, 4, ... digits is, respectively, 2, 11, 101, 1009, .... Here we list the sequence of digits of these numbers: 2: 1, 1; 1, 0, 1; 1, 0, 0, 9; ...
This can be considered, as for the Champernowne and Copeland-Erdős constants, as the decimal expansion of a real constant 0.2111011009...

Cf. A098449 (smallest n-digit semiprime), A215647 (has this as "limit"), A340206 (same for squares, limit of A215689), A340207 (similar, with largest n-digit squares, limit of A339978), A340208 (same for cubes, limit of A215692), A340209 (same with largest n-digit cube, limit of A340115), A340219 (same for primes, limit of A215641).

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

concat([digits(A098449(k))|k<-[1..14]]) \\ as seq. of digits
c(N=20)=sum(k=1,N,.1^(k*(k+1)/2)*A098449(k)) \\ as constant

c = 0.410106100310001100001100000110000001100000001100000000610000000003...
  = Sum_{k >= 1} 10^(-k(k+1)/2)*A098449(k)
a(-n(n+1)/2) = 1 for all n >= 2, followed by increasingly more zeros.

A378328 Decimal expansion of the base 4 Champernowne constant.

Original entry on oeis.org

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

Offset: 0

Joshua Searle, Nov 23 2024

This constant is formed by the concatenation of the natural numbers in base 4 and then converted into base 10.

This constant is 4-normal.

			0.426111111111111065764556571420161985095546238967230410682791635172587553...

OeisWiki, Champernowne constant

Cf. A030302, A003137, A030373, A031219, A030548, A030998, A054634, A031076, A033307 (base n expansions of base n Champernowne constants, without leading zero, for 2 <= n <= 10).
Cf. A066716, A077771, A378328, A378329, A378330, A378331, A378332, A378333, A033307 (decimal expansions of base n Champernowne constants for 2 <= n <= 10).
Cf. A066717, A077772, A378345, A378346, A378347, A378348, A378349, A378350, A030167 (continued fraction expansions of base n Champernowne constants for 2 <= n <= 10).

First[RealDigits[ChampernowneNumber[4], 10, 100]]

A378329 Decimal expansion of the base 5 Champernowne constant.

Original entry on oeis.org

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

Offset: 0

Joshua Searle, Nov 23 2024

This constant is formed by the concatenation of the natural numbers in base 5 and then converted into base 10.

This constant is 5-normal.

			0.310736111111111111111111111110963033311604944849115504682622268470343392...

OeisWiki, Champernowne constant

Cf. A030302, A003137, A030373, A031219, A030548, A030998, A054634, A031076, A033307 (base n expansions of base n Champernowne constants, without leading zero, for 2 <= n <= 10).
Cf. A066716, A077771, A378328, A378329, A378330, A378331, A378332, A378333, A033307 (decimal expansions of base n Champernowne constants for 2 <= n <= 10).
Cf. A066717, A077772, A378345, A378346, A378347, A378348, A378349, A378350, A030167 (continued fraction expansions of base n Champernowne constants for 2 <= n <= 10).

First[RealDigits[ChampernowneNumber[5], 10, 100]]

A378330 Decimal expansion of the base 6 Champernowne constant.

Original entry on oeis.org

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

Offset: 0

Joshua Searle, Nov 23 2024

This constant is formed by the concatenation of the natural numbers in base 6 and then converted into base 10.

This constant is 6-normal.

			0.239862685815066767447719828672209624590576971529350213760693195631576583...

OeisWiki, Champernowne constant

Cf. A030302, A003137, A030373, A031219, A030548, A030998, A054634, A031076, A033307 (base n expansions of base n Champernowne constants, without leading zero, for 2 <= n <= 10).
Cf. A066716, A077771, A378328, A378329, A378330, A378331, A378332, A378333, A033307 (decimal expansions of base n Champernowne constants for 2 <= n <= 10).
Cf. A066717, A077772, A378345, A378346, A378347, A378348, A378349, A378350, A030167 (continued fraction expansions of base n Champernowne constants for 2 <= n <= 10).

First[RealDigits[ChampernowneNumber[6], 10, 100]]

A378331 Decimal expansion of the base 7 Champernowne constant.

Original entry on oeis.org

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

Offset: 0

Joshua Searle, Nov 25 2024

This constant is formed by the concatenation of the natural numbers in base 7 and then converted into base 10.

This constant is 7-normal.

			0.194435535086240521475840093082908576452932971050422112479588531233679088...

OeisWiki, Champernowne constant

Cf. A030302, A003137, A030373, A031219, A030548, A030998, A054634, A031076, A033307
(base n expansions of base n Champernowne constants, without leading zero, for 2 <= n <= 10).
Cf. A066716, A077771, A378328, A378329, A378330, A378331, A378332, A378333, A033307 (decimal expansions of base n Champernowne constants for 2 <= n <= 10).
Cf. A066717, A077772, A378345, A378346, A378347, A378348, A378349, A378350, A030167 (continued fraction expansions of base n Champernowne constants for 2 <= n <= 10).

First[RealDigits[ChampernowneNumber[7], 10, 100]]

A378332 Decimal expansion of the base 8 Champernowne constant.

Original entry on oeis.org

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

Offset: 0

Joshua Searle, Nov 25 2024

This constant is formed by the concatenation of the natural numbers in base 8 and then converted into base 10.

This constant is 8-normal.

			0.163264812105216797367094986142605190224237843285462333081380700428319475...

OeisWiki, Champernowne constant

Cf. A030302, A003137, A030373, A031219, A030548, A030998, A054634, A031076, A033307 (base n expansions of base n Champernowne constants, without leading zero, for 2 <= n <= 10).
Cf. A066716, A077771, A378328, A378329, A378330, A378331, A378332, A378333, A033307 (decimal expansions of base n Champernowne constants for 2 <= n <= 10).
Cf. A066717, A077772, A378345, A378346, A378347, A378348, A378349, A378350, A030167 (continued fraction expansions of base n Champernowne constants for 2 <= n <= 10).

First[RealDigits[ChampernowneNumber[8], 10, 100]]

A378333 Decimal expansion of the base 9 Champernowne constant.

Original entry on oeis.org

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

Offset: 0

Joshua Searle, Nov 25 2024

This constant is formed by the concatenation of the natural numbers in base 9 and then converted into base 10.

This constant is 9-normal.

			0.140624976119696782479669008935663183265457083246828486657555171275414914...

OeisWiki, Champernowne constant

Cf. A030302, A003137, A030373, A031219, A030548, A030998, A054634, A031076, A033307 (base n expansions of base n Champernowne constants, without leading zero, for 2 <= n <= 10).
Cf. A066716, A077771, A378328, A378329, A378330, A378331, A378332, A378333, A033307 (decimal expansions of base n Champernowne constants for 2 <= n <= 10).
Cf. A066717, A077772, A378345, A378346, A378347, A378348, A378349, A378350, A030167 (continued fraction expansions of base n Champernowne constants for 2 <= n <= 10).

First[RealDigits[ChampernowneNumber[9], 10, 100]]

A132133 Number of n-digit "Punctual Birds" (cf. A131881).

Original entry on oeis.org

1, 9, 45, 270, 2104, 16941, 142245, 1226146

Offset: 0

Graeme McRae, Aug 11 2007

The number of n-digit "Early Birds" is 9*10^(n-1) - A132133(n), which is 0, 45, 630, 6896, 73059, 757755, 7773854, ... for n = 1, 2, ...

Here a(0) = 1 corresponds to the number 0 which is punctual in the sense that it does not occur before position 0, and which may be considered to have 0 digits for convenience: The index of 10^n in A131881 is then Sum_{k=0..n} a(k). - M. F. Hasler, Oct 25 2019

			a(2) = 45 because there are 45 2-digit Punctual Birds (10, 11, 13-20, 22, 24-30, 33, 35-40, 44, 46-50, 55, 57-60, 66, 68-70, 77, 79, 80, 88 and 90)

Cf. A033307, A117804, A116700, A131881, A132131, A132132.

Extended to a(0) = 1 by M. F. Hasler, Oct 25 2019

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

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;

cp's OEIS Frontend

A340219 Constant whose decimal expansion is the concatenation of the smallest n-digit prime A003617(n), for n = 1, 2, 3, ...

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A340221 Constant whose decimal expansion is the concatenation of the smallest n-digit semiprime A098449(n), for n = 1, 2, 3, ...

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A378328 Decimal expansion of the base 4 Champernowne constant.

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A378329 Decimal expansion of the base 5 Champernowne constant.

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A378330 Decimal expansion of the base 6 Champernowne constant.

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A378331 Decimal expansion of the base 7 Champernowne constant.

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A378332 Decimal expansion of the base 8 Champernowne constant.

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A378333 Decimal expansion of the base 9 Champernowne constant.

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