A218030 Numbers k equal to half of the product of the nonzero (base-10) digits of k^2.

2, 5, 54, 648, 2160, 337169025526136832000, 685506275314921762068267522458966662115416623590907309075726336000000, 46641846972427276691124922228108091690332947069125333309512419901440000000000

Offset: 1

Author

Nels Olson, Oct 18 2012

Keywords

• nonn
• base

Comments

The first 5 terms of the sequence were found by the author around 1980 using his Commodore PET computer. He found the subsequent terms in 1991 by means of an improved program. The author has always referred to these as the "Faithy numbers" after his mother, Faith, who posed the problem.

Examples

			For n=5, n^2 is 25; the product of the digits of 25 is 2*5 = 10, which is equal to 2*n.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..12 (all terms < 10^300)
• Michael S. Branicky, Python program.
• Giovanni Resta, C program for this and related sequences.

Crossrefs

Special case of A218013 where the ratio of the digit-product to the original number is 2. Related to A218072.

Programs

Mathematica

mx = 2^255; L = {};
p2 = 1; While[p2 < mx, Print["--> 2^", Log[2, p2]];
p3 = p2; While [p3 < mx,
  p5 = p3; While[p5 < mx,
   n = p5; While[n < mx,
    If[2 n == Times @@ Select[IntegerDigits[n^2], # > 0 &],
     AppendTo[L, n]; Print[n]]; n *= 7]; p5 *= 5]; p3 *= 3];
p2 *= 2]; Sort[L] (* Giovanni Resta, Oct 19 2012 *)

PARI

is_A218030(n)={my(d=digits(n^2));n*=2;for(i=1,#d,d[i]||next;n%d[i]&return;n\=d[i]);n==1} \\ M. F. Hasler, Oct 19 2012