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

%I A275971 #29 May 15 2021 10:25:16
%S A275971 5,15,85,165,235,485,1665,1885,4835,5765,7585,15085,15885,16665,18365,
%T A275971 18915,22885,27115,27885,50235,57665,58115,72335,85635,87885,150915,
%U A275971 166665,182415,194235,194365,229635,240365,268835,503515,507665,524915,568835,570415,577515,581165
%N A275971 Numbers n such that the decimal digits of n^2 are all prime.
%C A275971 Apparently 5, 235 and 72335 are the only terms using digits {2,3,5,7}.
%C A275971 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
%C A275971 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
%H A275971 Michel Marcus, <a href="/A275971/b275971.txt">Table of n, a(n) for n = 1..1000</a>
%F A275971 a(n) = sqrt(A191486(n)).
%e A275971 72335^2 = 5232352225 = A191486(23).
%t A275971 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 *)
%o A275971 (PARI) is(n)=#setintersect(Set(digits(n^2)), [0, 1, 4, 6, 8, 9])==0 \\ _Charles R Greathouse IV_, Sep 16 2016
%o A275971 (Python)
%o A275971 def aupto(limit):
%o A275971   alst = []
%o A275971   for k in range(1, limit+1):
%o A275971     if set(str(k*k)) <= set("2357"): alst.append(k)
%o A275971   return alst
%o A275971 print(aupto(10**6)) # _Michael S. Branicky_, May 15 2021
%Y A275971 Cf. A191486, A017329, A030487.
%K A275971 nonn,base
%O A275971 1,1
%A A275971 _Zak Seidov_, Aug 15 2016
%E A275971 More terms from _Michel Marcus_, Aug 17 2016