A340115 -id:A340115 - cp's OEIS frontend

A215692 Smallest 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.

0, 127, 127343, 1275122197, 127125100019683, 127125100012167148877, 1271251000106481038233442951, 127125100010648103823100000014348907, 127125100010648103823100000010077696108531333, 1271251000106481038231000000100776961005446251939096223

Offset: 1

Jonathan Vos Post, Aug 20 2012

This is to cubes A000578 as A215689 is to squares A000290.

The n-th term has A000217(n) = n(n+1)/2 digits. 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 smallest k digit cube, cf. formula. The number of such primes between a(n) and A340115(n) (the largest of this form) is (0, 2, 2, 9, 177, 6909, 570166, ...). (In particular, for n = 2 and 3, a(n) and A340115(n) are the only two primes of this form.) 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) = 127 because 127 is prime and is the concatenation of 1=1^3 and 27 = 3^3.

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

Cf. A000040 (primes), A000578 (cubes), A215689, A215641, A215647 (analog for squares, primes, semiprimes).

Cf. A340115 (largest prime of the given form), A000217 (triangular numbers: length of n-th term).

apply( {A215692(n)=forvec(v=vector(n,k,[ceil(10^((k-1)/3)),sqrtnint(10^k-1,3)]),ispseudoprime(n=eval(concat([Str(k^3)|k<-v])))&&return(n))}, [1..12]) \\ M. F. Hasler, Dec 31 2020

a(n) ~ 10^(n(n+1)/2)*0.1271251000106481038231000000100776961... (conjectured) - M. F. Hasler, Dec 31 2020

More terms (up to a(10)) from Alois P. Heinz, Aug 21 2012

A338968 a(n) is the largest prime whose decimal expansion consists of the concatenation of a 1-digit prime, a 2-digit prime, a 3-digit prime, ..., and an n-digit prime.

Original entry on oeis.org

7, 797, 797977, 7979979941, 797997997399817, 797997997399991999371, 7979979973999919999839999901, 797997997399991999983999999199999131, 797997997399991999983999999199999989999997639, 7979979973999919999839999991999999899999999379999997871

Offset: 1

Bernard Schott, Dec 21 2020

It is a plausible conjecture that a(n) always exists and begins with 7.
The similar smallest primes are in A215641.
If a(n) exists, it has A000217(n) = n*(n+1)/2 digits.
a(1) = 7 = A003618(1) and a(2) = 797 is the concatenation of 7 = A003618(1) and 97 = A003618(2)  that are respectively the largest 1-digit prime and 2-digit prime.
Conjecture: for n >= 3, a(n) is the concatenation of the largest k-digit primes with 1 <= k <= n-1: A003618(1)/A003618(2)/.../A003618(n-1) but the last concatenated prime with n digits is always < A003618(n). This conjecture has been checked by Daniel Suteu until a(360), a prime with 64980 digits.

			a(3) = 797977 is the largest prime formed from the concatenation of a single-digit, a double-digit, a triple-digit prime, i.e., 7, 97, 977.
a(4) = 7979979941 is the largest prime formed from the concatenation of a single-digit, a double-digit, a triple-digit, and a quadruple-digit prime, i.e., 7, 97, 997, 9941.

Cf. A000217, A003618, A215641.
Subsequence of A195302.
Cf. A339978 (with concatenated squares), A340115 (with concatenated cubes).

More terms from David A. Corneth, Dec 21 2020

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

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.

A340208 Constant whose decimal expansion is the concatenation of the smallest n-digit cube A061434(n), for n = 1, 2, 3, ...

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Dec 31 2020

Every third smallest n-digit cube (i.e., for n = 3k + 1, k >= 0) is 10^k, which explains the chunks of (1,0,...,0), cf. formula.

The terms of sequence A215692 converge to this sequence of digits, and to this constant, up to powers of 10.

			The smallest cube with 1, 2, 3, 4, ... digits is, respectively, 1, 27 = 3^3, 125 = 5^3, 1000 = 10^3, .... Here we list the sequence of digits of these numbers: 1; 2, 7; 1, 2, 5; 1, 0, 0, 0; ...
This can be considered, as for the Champernowne and Copeland-Erdős constants, as the decimal expansion of a real constant 0.1271251000106481...
As a triangle, in which row n contains the decimal expansion of the smallest n-digit cube:
  1
  2 7
  1 2 5
  1 0 0 0
  1 0 6 4 8
  1 0 3 8 2 3
  1 0 0 0 0 0 0
  1 0 0 7 7 6 9 6
  ...

Zhuorui He, Table of n, a(n) for n = 0..11324 (first 150 rows)

Cf. A061434 (smallest n-digit cube), A215692 (has this as "limit"), A340209 (same with largest n-digit cubes, limit of A340115), A340206 (same for squares, limit of A215689), A340219 (same for primes, limit of A215641), 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).

concat([digits(ceil(10^((k-1)/3))^3)|k<-[1..14]]) \\ as seq. of digits
c(N=12)=sum(k=1,N,.1^(k*(k+1)/2)*ceil(10^((k-1)/2))^2) \\ as constant

c = 0.12712510001064810382310000001007769610054462510000000001000787387510002657...
  = Sum_{k >= 1} 10^(-k(k+1)/2)*ceiling(10^((k-1)/3))^2
a(-n(n+1)/2) = 1 for all n >= 2;
a(k) = 0 for -3n(3n+1)/2 > k > -(3n+1)(3n+2)/2, n >= 0.

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

Original entry on oeis.org

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.

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;

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

M. F. Hasler, Jan 01 2021

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

			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...

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).

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

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

M. F. Hasler, Jan 01 2021

			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...

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).

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

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.

cp's OEIS Frontend

A215692 Smallest 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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A338968 a(n) is the largest prime whose decimal expansion consists of the concatenation of a 1-digit prime, a 2-digit prime, a 3-digit prime, ..., and an n-digit prime.

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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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A340208 Constant whose decimal expansion is the concatenation of the smallest n-digit cube A061434(n), for n = 1, 2, 3, ...

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

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