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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.

%I A215689 #17 Jan 01 2021 12:04:11
%S A215689 0,149,125441,1161002209,116100102414161,116100102410000106929,
%T A215689 1161001024100001004891442401,116100102410000100489100000010169721,
%U A215689 116100102410000100489100000010004569100460529,1161001024100001004891000000100045691000000001009269361
%N 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.
%C A215689 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
%H A215689 M. F. Hasler, <a href="/A215689/b215689.txt">Table of n, a(n) for n = 1..44</a> (all terms < 10^1000), Dec 31 2020.
%F A215689 a(n) ~ 10^(n(n+1)/2) * 0.1161001024100001004891000000100045691... - _M. F. Hasler_, Dec 31 2020
%e A215689 a(2) = 149, which is a prime, and the concatenation of 1 = 1^2 with 49 = 7^2.
%e A215689 a(3) = 125441, which is a prime, and the 1 = 1^2 with 25 = 5^2 with 441 = 21^2.
%o A215689 (PARI) 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
%Y A215689 Cf. A000040, A000290, A000217, A215641, A215647.
%Y A215689 Cf. A215692 (analog for cubes).
%K A215689 nonn,base
%O A215689 1,2
%A A215689 _Jonathan Vos Post_, Aug 20 2012
%E A215689 More terms (up to a(10)) from _Alois P. Heinz_, Aug 21 2012