A225885 Square numbers that remain square when their most-significant (or leftmost) digit is removed.

1, 4, 9, 49, 64, 81, 100, 225, 400, 625, 900, 1225, 2025, 3025, 4225, 4900, 5625, 6400, 7225, 8100, 9025, 10000, 15625, 22500, 27225, 30625, 34225, 40000, 42025, 50625, 60025, 62500, 70225, 75625, 81225, 90000, 93025, 105625, 122500, 202500, 275625, 302500, 330625

Offset: 1

Author

Kevin L. Schwartz and Christian N. K. Anderson, May 19 2013

Keywords

• nonn
• base

Comments

The first three are the only terms not divisible by 25 (and thus, not ending in 00 or 25). If 100 is a term, then the sequence should start with 3 more initial terms, namely (1, 4, 9, ...) - M. F. Hasler, Nov 01 2014

Examples

			225 = 15^2 becomes 25 = 5^2, 105625 = 325^2 becomes 5625 = 75^2.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..3000 (n = 4..403 from Christian N. K. Anderson and Kevin L. Schwartz)

Crossrefs

Cf. A225873, A249587, A247267.

Programs

Mathematica

b^2 /. Flatten[Outer[Solve[a^2 + #2*10^#1 == b^2 && 0 <= a < Sqrt[10^#1] && Sqrt[#2*10^#1] <= b < Sqrt[10^(#1 + 1)], {a, b}, Integers] &, Range[0, 5], Range[9]], 2] (* Davin Park, Dec 30 2016 *)

PARI

is_A225885(n)=issquare(n%10^(#Str(n)-1))&&issquare(n)&&n>9 \\ M. F. Hasler, Nov 01 2014

R

no0<-function(s){ while(substr(s,1,1)=="0" & nchar(s)>1) s=substr(s,2,nchar(s)); s};
issquare<-function(x) ifelse(as.bigz(x)<2,T,all(table(as.numeric(gmp::factorize(x)))%%2==0));
which(sapply(1:200,function(x) issquare(no0(substr(x^2,2,ndig(x^2)))))>0)^2

Extensions

1,4,9 added (per M. F. Hasler's comment) by Chai Wah Wu, Nov 03 2014