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

1,1

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

Links

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

Formula

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

Example

72335^2 = 5232352225 = A191486(23).

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 *)

Prog

(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

Crossrefs

Cf. A191486, A017329, A030487.

Sequence in context: A064678 A088935 A183937 * A030487 A165470 A165625

Adjacent sequences: A275968 A275969 A275970 * A275972 A275973 A275974

Keyword

nonn,base

Author

Zak Seidov, Aug 15 2016

Extensions

More terms from Michel Marcus, Aug 17 2016

Status

approved