A191486 Squares using only the prime digits (2,3,5,7).

25, 225, 7225, 27225, 55225, 235225, 2772225, 3553225, 23377225, 33235225, 57532225, 227557225, 252333225, 277722225, 337273225, 357777225, 523723225, 735223225, 777573225, 2523555225, 3325252225, 3377353225, 5232352225, 7333353225

Offset: 1

Author

Giovanni Teofilatto, Jun 03 2011

Keywords

• nonn
• base

Comments

a(n) = 225 mod 1000 for n > 1. - Charles R Greathouse IV, May 14 2013

The sequence is infinite: it contains A030485 as an infinite proper subsequence which in turn contains all numbers of the form ((5*10^n-5)/3)^2 as a proper subsequence. - M. F. Hasler, Sep 16 2016

Links

• Charles R Greathouse IV and Donovan Johnson, Table of n, a(n) for n = 1..1000 (first 155 terms from Charles R Greathouse IV)

Crossrefs

Cf. A077676, A030485.

Programs

Magma

[n^2: n in [5..5*10^5] | Set(Intseq(n^2)) subset {2,3,5,7}];  // Bruno Berselli, Jun 06 2011

Maple

for b from 1 do convert(convert(b^2,base,10),set) ; if % minus {2,3,5,7} = {} then printf("%d,\n",b^2) ; end if; end do: # R. J. Mathar, Jun 03 2011

Mathematica

w = Boole@! PrimeQ@ # & /@ RotateLeft@ Range[0, 9]; Select[Range[10^5]^2, Total@ Pick[DigitCount@ #, w, 1] == 0 &] (* Michael De Vlieger, Aug 15 2016 *)

PARI

toprime(n,k)=n<<=2;sum(i=0,k-1,n>>=2;[2,3,5,7][bitand(n,3)+1]*10^i)
v=List([25]);for(k=0,9,for(n=0,4^k-1,t=1000*toprime(n,k)+225;if(issquare(t),listput(v,t)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, May 14 2013

Python

from math import isqrt
def aupto(limit):
  alst, rootlimit = [], isqrt(limit)
  for k in range(1, rootlimit+1):
    if set(str(k*k)) <= set("2357"): alst.append(k*k)
  return alst
print(aupto(7333353225)) # Michael S. Branicky, May 15 2021

Formula

a(n) = A275971(n)^2. - M. F. Hasler, Sep 16 2016