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.
A124809(1)=50, 50 = 2*5*5, so 50 is not squarefree, the square being 5*5 = 25.
A124809(1)=50, 50 = 2*5*5, so 50 is not squarefree, the square being 5*5 = 25 and the root being 5.
a(1) = 7 because 7^2 + 1 = 49 + 1 = 50 is divisible by 25, a square.
[n: n in [1..6*10^2]| not IsSquarefree(n^2+1)]; // Bruno Berselli, Oct 15 2012
n=1;Reap[Do[While[SquareFreeQ[n^2+1],n++];Sow[n];n++,{c,10000}]][[2,1]] (* Zak Seidov, Feb 24 2011 *)
for(n=1,1e4,if(!issquarefree(n^2+1),print1(n", "))) \\ Charles R Greathouse IV, Feb 24 2011
select(numtheory:-issqrfree, [seq(n^2+1, n=1..100)]); # Robert Israel, Feb 09 2016
Select[ Range[10^4], IntegerQ[ Sqrt[ # - 1]] && Union[ Transpose[ FactorInteger[ # ]] [[2]]] [[ -1]] == 1 &] Select[Range[60]^2+1,SquareFreeQ] (* Harvey P. Dale, Mar 21 2013 *)
for(n=1,100,if(issquarefree(n^2+1),print1(n^2+1,",")))
a(1)=10 because 10 = 3^2 + 1 is squarefree. a(2)=26 because 26 = 5^2 + 1 is squarefree. a(3)=65 because 65 = 8^2 + 1 is squarefree.
ts_fn3:=proc(n) local i,tren,ans; ans:=[ ]: for i from 1 to n do tren := i^(2)+1: if (isprime(tren) = false and numtheory[mobius] (tren) <> 0 ) then ans:=[ op(ans), tren ]: fi od: RETURN(ans) end: ts_fn3(200);
Select[Range[70]^2+1, CompositeQ[#] && SquareFreeQ[#] &] (* Amiram Eldar, Feb 22 2021 *)
a(1)=3, because 3^2 + 1 = 10 is composite squarefree. a(2)=5, because 5^2 + 1 = 26 is composite squarefree. a(3)=8, because 8^2 + 1 = 50 is composite squarefree.
ts_fn4:=proc(n) local i,tren,ans; ans:=[ ]: for i from 1 to n do tren := i^(2)+1: if (isprime(tren) = false and numtheory[mobius] (tren) <> 0 ) then ans:=[ op(ans), i ]: fi od: RETURN(ans) end: ts_fn4(200);
Select[Range[100], CompositeQ[#^2+1] && SquareFreeQ[#^2+1] &] (* Amiram Eldar, Feb 22 2021 *)
7 is in the sequence because of the pair (m, k) = (7, 3), 7^2+1 = 2*5^2 and 3^2+1 = 2*5 with the same prime factors 2 and 5.
Select[Range@ 5000, Function[m, Total@ Boole@ Table[Function[w, And[SameQ[First@ w, #], SameQ[Last@ w, #]] &@ Union@ Flatten@ w]@ Map[FactorInteger[#][[All, 1]] &, {m^2 + 1, k^2 + 1}], {k, m - 1}] > 0]] (* Michael De Vlieger, Feb 07 2017 *)
isok(n)=ok = 0; vn = factor(n^2+1)[,1]; for (k=1, n-1, if (factor(k^2+1)[,1] == vn, ok = 1; break);); ok; \\ Michel Marcus, Feb 09 2017
squeeze(f)=factorback(f)\2 list(lim)=my(v=List(),m=Map(),t); for(n=1,lim, t=squeeze(factor(n^2+1)[,1]); if(mapisdefined(m,t), listput(v,n), mapput(m,t,0))); Vec(v) \\ Charles R Greathouse IV, Feb 12 2017
use ntheory qw(:all);
for (my ($m, %t) = 1 ; ; ++$m) {
my $k = vecprod(map{$_->[0]}factor_exp($m**2+1));
push @{$t{$k}}, $m;
if (@{$t{$k}} >= 2) {
print'('.join(', ',reverse(@{$t{$k}})).")\n";
}
} # Daniel Suteu, Feb 08 2017
Comments