A195985 Least prime such that p^2 is a zeroless n-digit number.

2, 5, 11, 37, 107, 337, 1061, 3343, 10559, 33343, 105517, 333337, 1054133, 3333373, 10540931, 33333359, 105409309, 333333361, 1054092869, 3333333413, 10540925639, 33333333343, 105409255363, 333333333367, 1054092553583, 3333333333383, 10540925534207

Offset

1,1

Links

M. F. Hasler, Table of n, a(n) for n = 1..158

C. Rivera, Puzzle 607. A zeroless Prime power, on primepuzzles.net, Sept. 24, 2011.

W. Schneider, NoZeros: Powers n^k without Digit Zero (local copy of www.wschnei.de/digit-related-numbers/nozeros.html), as of Jan 30 2003.

Example

a(1)^2=4, a(2)^2=25, a(3)^2=121, a(4)^2=1369 are the least squares of primes with 1, 2, 3 resp. 4 digits, and these digits are all nonzero.
a(5)=107 since 101^2=10201 and 103^2=10609 both contain a zero digit, but 107^2=11449 does not.
a(1000)=[10^500/3]+10210 (500 digits), since primes below sqrt(10^999) = 10^499*sqrt(10) ~ 3.162e499 have squares of less than 1000 digits, between sqrt(10^999) and 10^500/3 = sqrt(10^1000/9) ~ 3.333...e499 they have at least one zero digit. Finally, the 7 primes between 10^500/3 and a(1000) also happen to have a "0" digit in their square, but not so
  a(1000)^2 = 11111...11111791755555...55555659792849
  = [10^500/9]*(10^500+5) + 6806*10^500+104237294.

Prog

(PARI) a(n)={ my(p=sqrtint(10^n\9)-1); until( is_A052382(p^2), p=nextprime(p+2)); p}

Crossrefs

Cf. A195942, A195943, A195944, A195945, A195946, A195908, A195948, A007377, A008839, A030700, A030701, A030702, A030703, A030704, A030705, A030706.

Sequence in context: A092298 A074497 A131581 * A367969 A056301 A001344

Adjacent sequences: A195982 A195983 A195984 * A195986 A195987 A195988

Keyword

nonn,base

Author

M. F. Hasler, Sep 26 2011

Status

approved