

A215689


Smallest prime whose decimal expansion consists of the concatenation of a 1digit square, a 2digit square, a 3digit square, ..., and an ndigit square, or 0 if there is no such prime.


8



0, 149, 125441, 1161002209, 116100102414161, 116100102410000106929, 1161001024100001004891442401, 116100102410000100489100000010169721, 116100102410000100489100000010004569100460529, 1161001024100001004891000000100045691000000001009269361
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OFFSET

1,2


COMMENTS

The nth 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 ndigit square, cf. formula.  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.


FORMULA

a(n) ~ 10^(n(n+1)/2) * 0.1161001024100001004891000000100045691...  M. F. Hasler, Dec 31 2020


EXAMPLE

a(2) = 149, which is a prime, and the concatenation of 1 = 1^2 with 49 = 7^2.
a(3) = 125441, which is a prime, and the 1 = 1^2 with 25 = 5^2 with 441 = 21^2.


PROG

(PARI) apply( {A215689(n)=forvec(v=vector(n, k, [ceil(10^((k1)/2)), sqrtint(10^k1)]), ispseudoprime(n=eval(concat([Str(k^2)k<v])))&&return(n))}, [1..11]) \\ M. F. Hasler, Dec 31 2020


CROSSREFS

Cf. A000040, A000290, A000217, A215641, A215647.
Cf. A215692 (analog for cubes).
Sequence in context: A188566 A188753 A188873 * A104262 A048706 A054729
Adjacent sequences: A215686 A215687 A215688 * A215690 A215691 A215692


KEYWORD

nonn,base


AUTHOR

Jonathan Vos Post, Aug 20 2012


EXTENSIONS

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


STATUS

approved



