This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
9605_9601 = 9801^2.
902406_902401 = 949951^2.
4023943_4023936 = 6343456^2.
638370_638361 = 798981^2.
fQ[n_] := IntegerQ[Sqrt[n*10^Floor[1 + Log10[n + 1]] + n + 1]]; (* Robert G. Wilson v, Dec 27 2017 *)
lista(nn) = forstep(n=183, nn, [3, 5, 7, 5, 3, 1, 4, 7, 5, 3, 5, 7, 5, 3, 5, 7, 5, 3, 5, 7, 4, 1], my(s = eval(concat(Str(n), Str(n+1)))); if(issquare(s), print1(s, ", "))) \\ Iain Fox, Dec 27 2017
eea(x, y) = my(a=max(x,y), b=min(x,y), s=0, so=1, st, r=b, ro=a, rt, q, t); while(r, q=ro\r; rt=r; r=ro-q*r; ro=rt; st=s; s=so-q*s; so=st); t=(ro-so*a)\b; if(x>y, [so, t], [t, so]) \\ Extended Euclidean Algorithm lista(nn) = my(res=Set(), b, f2, c, s); for(d=3, nn, b=10^d+1; fordiv(b, f, if(f!=1 && f!=b, f2=b/f; if(gcd(f, f2)==1, c=eea(f, f2); if(c[1]<0, s=f*(f2+2*c[1])*f2*(f-2*c[2])+1, s=f*(2*c[1])*f2*(-2*c[2])+1); if(#digits(s)==d*2, res=setunion(res, Set(s))))))); Vec(res) \\ (Will find all values of length nn*2 or shorter) Iain Fox, Oct 16 2021
'8242' + '8242-1' gives 82428241 which is 9079^2. Leading zeros are not allowed, which is why c(1)=266327 is not in this sequence although c(1)^2 = 070930 070929.
isA054216(n)={ 1==[1,-1]*divrem(n^2,10^(#Str(n^2)\2)) & #Str(n^2)%2==0 }
E.g. '8242' + '8242-1' gives 82428241 which is 9079^2.
The first few squares in binary are 1, 100, 1001, 10000, 11001, 100100. Thus we see that 100100, which is 36 in decimal, the square of 6, is the first square which is the concatenation of two identical bit patterns, and therefore a(2) = 6.
f:= proc(b)
local d,F,x,t,j;
for d from 1 do
F:= select(t -> t[2]::odd, ifactors(1+b^d)[2]);
x:= mul(t[1],t=F);
if x >= b^d then next fi;
j:= ceil(sqrt(b^(d-1)/x));
if j^2*x < b^d then return j*sqrt(x*(1+b^d)) fi
od
end proc:
map(f, [$2..40]); # Robert Israel, May 19 2024
a(11) = 704 because 704^2 = 495616 is the concatenation of 495 and 616, and 616 - 495 = 121 = 11^2.
for n from 1 to 47 do:
ii:=0:
for k from 1 to 10^7 while(ii=0)do :
x:=convert(k^2,base,10):n1:=nops(x):
if irem(n1,2)=0
then
s:=sum('x[i]*10^(i-1) ', 'i'=1..n1/2):
z:=convert(s,base,10):
s1:=sum('x[j]*10^(j-n1/2-1) ', 'j'=n1/2+1..n1):
if s-s1 = n^2
then
ii:=1:printf(`%d, `,k):
else
fi:
fi:
od:
od:
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