A117678 Squares for which the multiplicative digital root is also a square.

0, 1, 4, 9, 25, 100, 169, 196, 225, 256, 400, 529, 576, 625, 676, 900, 961, 1024, 1089, 1156, 1225, 1296, 1521, 1600, 2025, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 2916, 3025, 3136, 3481, 3600, 3844, 3969, 4096, 4225, 4356, 4489, 4900, 5041, 5184, 5329

Offset: 1

Author

Luc Stevens (lms022(AT)yahoo.com), Apr 12 2006

Keywords

• nonn
• easy
• base

Comments

From Robert Israel, Oct 22 2015: (Start)

1, 9, and squares in A034048 and A034051.

Are there infinitely many squares in A034051? (End)

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A000290, A031347, A034048, A034051, A116978.

Programs

Maple

A007954 := proc(n) return mul(d, d=convert(n, base, 10)): end: A117678 := proc(n) option remember: local k, m: if(n=1)then return 0:fi: for k from procname(n-1)+1 do m:=k^2: while(length(m)>1)do m:=A007954(m): od: if(m in {0,1,4,9})then return k: fi: od: end: seq(A117678(n)^2, n=1..47); # Nathaniel Johnston, May 05 2011

Mathematica

Select[Range[0, 73]^2, IntegerQ@ Sqrt[FixedPoint[Times @@ IntegerDigits@ # &, #] &@ #] &] (* Michael De Vlieger, Oct 22 2015 *)

PARI

t(k) = {while(k>9, k=prod(i=1, #k=digits(k), k[i])); k}
for(n=0, 100, if(issquare(t(n^2)), print1(n^2, ", "))); \\ Altug Alkan, Oct 22 2015

Extensions

Offset and some terms corrected by Nathaniel Johnston, May 05 2011