A249587 -id:A249587 - cp's OEIS frontend

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

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

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

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

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

Cf. A225873, A249587, A247267.

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 *)

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

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

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

A249853 Numbers whose cubes become squares if one of their digits is deleted.

Original entry on oeis.org

4, 5, 6, 10, 20, 21, 25, 40, 44, 64, 90, 100, 129, 160, 200, 250, 360, 400, 490, 500, 600, 640, 810, 1000, 1210, 1440, 1690, 1960, 2000, 2025, 2100, 2250, 2500, 2560, 2890, 3240, 3610, 4000, 4400, 4410, 4840, 5025, 5290, 5760, 6250, 6400, 6760, 7290, 7840, 8410

Offset: 1

Paolo P. Lava, Nov 07 2014

A245096 gives the numbers whose squares become cubes if one of their digit is deleted.

Numbers with single-digit cubes are not included. - Davin Park, Dec 30 2016

			21^3 = 9261 and sqrt(961) = 31.
44^3 = 85184 and sqrt(5184) = 72.
45625^3 = 94974853515625 and sqrt(9474853515625) = 3078125.

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 100 terms from Paolo P. Lava)

Cf. A249587, A225885.

with(numtheory): P:=proc(q) local a,k,n;
for n from 1 to q do a:=n^3; for k from 1 to ilog10(a) do
if type(sqrt(trunc(a/10^(k+1))*10^k+(a mod 10^k)),integer)
then print(n); break; fi; od; od; end: P(10^9);

f[n_] := !MissingQ@SelectFirst[Delete[IntegerDigits[n^3], #] & /@ Range[IntegerLength[n^3]], IntegerQ@Sqrt@FromDigits@# &];
Select[Range[4, 1000], f] (* Davin Park, Dec 30 2016 *)

A247267 Square numbers not divisible by 100 that remain square when their most-significant (or leftmost) digit is removed.

Original entry on oeis.org

1, 4, 9, 49, 64, 81, 225, 625, 1225, 2025, 3025, 4225, 5625, 7225, 9025, 15625, 27225, 30625, 34225, 42025, 50625, 60025, 70225, 75625, 81225, 93025, 105625, 275625, 330625, 600625, 893025, 950625, 970225, 1050625, 2030625, 3330625, 4950625, 9455625, 9765625, 15405625

Offset: 1

Chai Wah Wu, Nov 01 2014

Subsequence of A225885. It is easy to see that the multiples of 100 in A225885 are earlier entries of the sequence multiplied by 100, so this sequence removes this redundancy in some sense. It appears that all entries in A225885 after the first 3 entries ends in either 25 or 00, so this sequence appears to end in 25 after the first 3 entries.

Chai Wah Wu, Table of n, a(n) for n = 1..1185

Cf. A225885, A247885. A249587.

for(n=10,10^6,if(n%100,if(issquare(n)&&issquare(n%(10^(#Str(n)-1))),print1(n,", ")))) \\ Derek Orr, Nov 01 2014

Added 1, 4, 9 to sequence to match A225885 - Chai Wah Wu, Nov 03 2014

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

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A249853 Numbers whose cubes become squares if one of their digits is deleted.

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A247267 Square numbers not divisible by 100 that remain square when their most-significant (or leftmost) digit is removed.

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