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.

%I A215692 #11 Jan 01 2021 12:04:31
%S A215692 0,127,127343,1275122197,127125100019683,127125100012167148877,
%T A215692 1271251000106481038233442951,127125100010648103823100000014348907,
%U A215692 127125100010648103823100000010077696108531333,1271251000106481038231000000100776961005446251939096223
%N 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.
%C A215692 This is to cubes A000578 as A215689 is to squares A000290.
%C A215692 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
%H A215692 M. F. Hasler, <a href="/A215692/b215692.txt">Table of n, a(n) for n = 1..44</a> (all terms < 10^1000), Dec 31 2020.
%F A215692 a(n) ~ 10^(n(n+1)/2)*0.1271251000106481038231000000100776961... (conjectured) - _M. F. Hasler_, Dec 31 2020
%e A215692 a(1) = 0 because no 1-digit cube {0,1,8} is prime.
%e A215692 a(2) = 127 because 127 is prime and is the concatenation of 1=1^3 and 27 = 3^3.
%o A215692 (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
%Y A215692 Cf. A000040 (primes), A000578 (cubes), A215689, A215641, A215647 (analog for squares, primes, semiprimes).
%Y A215692 Cf. A340115 (largest prime of the given form), A000217 (triangular numbers: length of n-th term).
%K A215692 nonn,base
%O A215692 1,2
%A A215692 _Jonathan Vos Post_, Aug 20 2012
%E A215692 More terms (up to a(10)) from _Alois P. Heinz_, Aug 21 2012