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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. 10

%I

%S 0,827,164729,8642164913,864729685979507,864729926197336531441,

%T 8647299261973369702994826809,864729926197336970299980034443986977,

%U 864729926197336970299993837599897344909853209,8647299261973369702999938375998973449970029998036054027

%N 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.

%C If a(n) exists it has A000217(n) = n*(n+1)/2 digits.

%C The similar smallest primes are in A215692.

%C 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

%H M. F. Hasler, <a href="/A340115/b340115.txt">Table of n, a(n) for n = 1..44</a> (all terms < 10^1000), Dec 31 2020

%e a(1) = 0 because no 1-digit cube {0, 1, 8} is prime.

%e a(2) = 827 because 827 is prime and is the concatenation of 8 = 2^3 and 27 = 3^3.

%e 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.

%o (Python)

%o from sympy import isprime

%o from itertools import product

%o def a(n):

%o cubes = [str(k**3) for k in range(1, int((10**n)**(1/3))+2)]

%o revcbs = [[k3 for k3 in cubes if len(k3)==i+1][::-1] for i in range(n)]

%o for t in product(*revcbs):

%o intt = int("".join(t))

%o if isprime(intt): return intt

%o return 0

%o print([a(n) for n in range(1, 11)]) # _Michael S. Branicky_, Dec 28 2020

%o (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

%Y Cf. A000217, A000578, A003618, A061435.

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

%K nonn,base

%O 1,2

%A _Bernard Schott_, Dec 28 2020

%E a(4)-a(10) from _Michael S. Branicky_, Dec 28 2020

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Last modified July 25 03:36 EDT 2021. Contains 346282 sequences. (Running on oeis4.)