A215692 -id:A215692 - cp's OEIS frontend

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.

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

A215689 Smallest prime whose decimal expansion consists of the concatenation of a 1-digit square, a 2-digit square, a 3-digit square, ..., and an n-digit square, or 0 if there is no such prime.

Original entry on oeis.org

0, 149, 125441, 1161002209, 116100102414161, 116100102410000106929, 1161001024100001004891442401, 116100102410000100489100000010169721, 116100102410000100489100000010004569100460529, 1161001024100001004891000000100045691000000001009269361

Offset: 1

Jonathan Vos Post, Aug 20 2012

The n-th term has n(n+1)/2 digits (cf. A000217). There are (0, 3, 29, 991, 175210, ...) primes of that form, for n = 1, 2, 3, .... We can conjecture that a(n) > 0 for all n, and even that the terms converge to the concatenation of (s(1), s(2), s(3), ...) where s(n) is the smallest n-digit square, cf. formula. - M. F. Hasler, Dec 31 2020

			a(2) = 149, which is a prime, and the concatenation of 1 = 1^2 with 49 = 7^2.
a(3) = 125441, which is a prime, and the 1 = 1^2 with 25 = 5^2 with 441 = 21^2.

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

Cf. A000040, A000290, A000217, A215641, A215647.

Cf. A215692 (analog for cubes).

apply( {A215689(n)=forvec(v=vector(n, k, [ceil(10^((k-1)/2)), sqrtint(10^k-1)]), ispseudoprime(n=eval(concat([Str(k^2)|k<-v])))&&return(n))}, [1..11]) \\ M. F. Hasler, Dec 31 2020

a(n) ~ 10^(n(n+1)/2) * 0.1161001024100001004891000000100045691... - M. F. Hasler, Dec 31 2020

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

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

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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A215689 Smallest prime whose decimal expansion consists of the concatenation of a 1-digit square, a 2-digit square, a 3-digit square, ..., and an n-digit square, or 0 if there is no such 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, ...