A275971 Numbers n such that the decimal digits of n^2 are all prime.

5, 15, 85, 165, 235, 485, 1665, 1885, 4835, 5765, 7585, 15085, 15885, 16665, 18365, 18915, 22885, 27115, 27885, 50235, 57665, 58115, 72335, 85635, 87885, 150915, 166665, 182415, 194235, 194365, 229635, 240365, 268835, 503515, 507665, 524915, 568835, 570415, 577515, 581165

Offset: 1

Author

Zak Seidov, Aug 15 2016

Keywords

• nonn
• base

Comments

Apparently 5, 235 and 72335 are the only terms using digits {2,3,5,7}.

a(n)/5 = {1, 3, 17, 33, 47, 97, 333, 377, 967, 1153, 1517, 3017, 3177, 3333, ...}; terms b(n) that have n 3's must be in the sequence since (5 b(n))^2 yields the decimal number 2 followed by (n-1) 7's then n 2's, and ending in 5 (i.e., 225, 27225, 2772225). Thus 5 b(n) = {15, 165, 1665, 16665, etc.} appears in this sequence. - Michael De Vlieger, Aug 15 2016

All terms are odd multiples of 5 (A017329), i.e., must end in 5, which is the only digit whose square ends in a prime digit. The sequence contains A030487 as an infinite proper subsequence which in turn contains all numbers of the form (5*10^n-5)/3 (these are the above 5 b(n)) as a proper subsequence. - M. F. Hasler, Sep 16 2016

Examples

			72335^2 = 5232352225 = A191486(23).
		

Links

• Michel Marcus, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A191486, A017329, A030487.

Programs

Mathematica

w = Boole@! PrimeQ@ # & /@ RotateLeft@ Range[0, 9]; Sqrt@ Select[Range[10^6]^2, Total@ Pick[DigitCount@ #, w, 1] == 0 &] (* Michael De Vlieger, Aug 15 2016 *)

PARI

is(n)=#setintersect(Set(digits(n^2)), [0, 1, 4, 6, 8, 9])==0 \\ Charles R Greathouse IV, Sep 16 2016

Python

def aupto(limit):
  alst = []
  for k in range(1, limit+1):
    if set(str(k*k)) <= set("2357"): alst.append(k)
  return alst
print(aupto(10**6)) # Michael S. Branicky, May 15 2021

Formula

a(n) = sqrt(A191486(n)).

Extensions

More terms from Michel Marcus, Aug 17 2016