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

1,2

Comments

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

Links

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

Example

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.

Prog

(Python)
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
(PARI) 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

Crossrefs

Cf. A000217, A000578, A003618, A061435.

Cf. A338968 (with concatenated primes), A339978 (with concatenated squares).

Sequence in context: A102350 A108830 A341272 * A134726 A104281 A217551

Adjacent sequences: A340112 A340113 A340114 * A340116 A340117 A340118

Keyword

nonn,base

Author

Bernard Schott, Dec 28 2020

Extensions

a(4)-a(10) from Michael S. Branicky, Dec 28 2020

Status

approved