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

1,2

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

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)

Example

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

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 *)

Prog

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

Crossrefs

Cf. A225873, A249587, A247267.

Sequence in context: A327699 A013495 A164839 * A247267 A115687 A115711

Adjacent sequences: A225882 A225883 A225884 * A225886 A225887 A225888

Keyword

nonn,base

Author

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

Extensions

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

Status

approved