A180313 A sequence a(n) such that a(n+1)^2 - a(n)^2 are perfect squares.

3, 5, 13, 85, 221, 1445, 3757, 24565, 63869, 417605, 1085773, 7099285, 18458141, 120687845, 313788397, 2051693365, 5334402749, 34878787205, 90684846733, 592939382485, 1541642394461, 10079969502245, 26207920705837, 171359481538165, 445534651999229, 2913111186148805

Offset

1,1

Comments

The lexically smallest sequence with a(n+1)^2-a(n)^2 representing perfect squares is A018928.

This version here is constructed via a(n+1) = a(n)* sqrt( 1+((p^2-1)/(2p))^2) where p = A020639(a(n)) is the smallest prime divisor of the previous term.

Links

Table of n, a(n) for n=1..26.

Example

After a(1)=3, p=3 (again) and a(2) = 3*sqrt(1+ (8/6)^2) = 5.
After a(4)=85, p=5 and a(5) = 85*sqrt(1+ (24/10)^2) = 85*sqrt(169/25) = 221.

Maple

A020639 := proc(n) min(op(numtheory[factorset](n))) ; end proc:
A180313 := proc(n) option remember; if n = 1 then 3; else aprev := procname(n-1) ; p := A020639(aprev) ; aprev* sqrt(1+((p^2-1)/2/p)^2) ; end if; end proc:
for n from 1 to 30 do printf("%d, ", A180313(n)) ; end do: # R. J. Mathar, Sep 23 2010

Mathematica

spd[n_] := FactorInteger[n][[1, 1]];
a[n_] := a[n] = If[n == 1, 3, aprev = a[n-1];
    p = spd[aprev]; aprev*Sqrt[1+((p^2-1)/2/p)^2]];
Table[a[n], {n, 1, 26}] (* Jean-François Alcover, Feb 28 2024, after R. J. Mathar *)

Prog

(Perl) # use 5.12.0; use warnings; use Math::Prime::TiedArray; tie my @primes, 'Math::Prime::TiedArray';
sub SmallestPrimeDivisor ($) { my ($n) = @_; for my $p (@primes) { if ($n % $p == 0) { return $p; } } }
sub FindIncrement ($) { my ($n) = @_; my $p = SmallestPrimeDivisor $n; my $k = $n / $p; return $k * ($p ** 2 - 1) / 2; }
my $n = 3; say $n; for my $i (0 .. 23) { my $d = FindIncrement $n; $n = sqrt($d ** 2 + $n ** 2); say $n; }

Crossrefs

Sequence in context: A268021 A018928 A239381 * A053630 A155012 A121533

Adjacent sequences: A180310 A180311 A180312 * A180314 A180315 A180316

Keyword

nonn

Author

Valentin Tiriac (valtron2000(AT)gmail.com), Aug 26 2010

Extensions

Nomenclature normalized by R. J. Mathar, Sep 23 2010

Corrected indexing error introduced with previous edit - R. J. Mathar, Oct 01 2010

Status

approved