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

Author

Jonathan Vos Post, Aug 20 2012

Keywords

• nonn
• base

Comments

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

Examples

			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.
		

Links

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

Crossrefs

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).

Programs

PARI

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

Formula

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

Extensions

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