Divinacci Sequence

Question

Divinacci (OEIS)

Perform the Fibonacci sequence but instead of using:

f(n) = f(n-1)+f(n-2)

Use:

f(n) = sum(divisors(f(n-1))) + sum(divisors(f(n-2)))

For an input of n, output the nth term, your program should only have 1 input.

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First 14 terms (0-indexed, you may 1-index; state which you used):

0  | 0     # Initial               | []
1  | 1     # Initial               | [1] => 1
2  | 1     # [] + [1]              | [1] => 1
3  | 2     # [1] + [1]             | [1,2] => 3
4  | 4     # [1] + [1,2]           | [1,2,4] => 7
5  | 10    # [1,2] + [1,2,4]       | [1,2,5,10] => 18
6  | 25    # [1,2,4] + [1,2,5,10]  | [1,5,25] => 31
7  | 49    # [1,2,5,10] + [1,5,25] | [1,7,49] => 57
8  | 88    # [1,5,25] + [1,7,49]   | [1, 2, 4, 8, 11, 22, 44, 88] => 180
9  | 237   # [1,7,49] + [180]      | [1, 3, 79, 237] => 320
10 | 500   # [180] + [320]         | [1, 2, 4, 5, 10, 20, 25, 50, 100, 125, 250, 500] => 1092
11 | 1412  # [320] + [1092]        | [1, 2, 4, 353, 706, 1412] => 2478
12 | 3570  # [1092] + [2478]       | [1, 2, 3, 5, 6, 7, 10, 14, 15, 17, 21, 30, 34, 35, 42, 51, 70, 85, 102, 105, 119, 170, 210, 238, 255, 357, 510, 595, 714, 1190, 1785, 3570] => 10368
13 | 12846 # [2478] + [10368]      | [1, 2, 3, 6, 2141, 4282, 6423, 12846] => 25704
Etc...

You may choose whether or not to include the leading 0. For those who do: the divisors of 0 are [] for the purpose of this challenge.

It's code-golf lowest byte-count wins...

Answer 1

05AB1E, 9 bytes

XÎFDŠ‚ÑOO

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Explanation

XÎ          # initialize stack with 1,0,input
  F         # input times do
   D        # duplicate
    Š       # move down 2 places on the stack
     ‚      # pair the top 2 elements on the stack
      Ñ     # compute divisors of each
       OO   # sum twice

Answer 2

Mathematica, 45 40 bytes

If[#<3,1,Tr@Divisors@#0[#-i]~Sum~{i,2}]&

Mathematica's divisor related functions Divisors, DivisorSum and DivisorSigma are all undefined for n = 0 (rightly so), so we start from f(1) = f(2) = 1 and don't support input 0.

Defining it as an operator instead of using an unnamed function seems to be two bytes longer:

±1=±2=1
±n_:=Sum[Tr@Divisors@±(n-i),{i,2}]

Answer 3

Perl 6, 58 bytes

{my&d={sum grep $_%%*,1..$_};(0,1,{d($^a)+d $^b}...*)[$_]}

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Answer 4

Haskell, 55 bytes

f n=sum[a|n>1,k<-f<$>[n-1,n-2],a<-[1..k],mod k a<1]+0^n

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One-indexed.

Answer 5

Jelly, 10 9 bytes

ð,ÆDẎSð¡1

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Thanks to Dennis for -1.

Answer 6

Python 3, 88 83 81 bytes

f=lambda n:+(n<3)or g(f(n-1))+g(f(n-2))
g=lambda n,i=1:n>=i and(n%i<1)*i+g(n,i+1)

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Excludes the 0

Answer 7

Python 2, 76 bytes

f=lambda n:sum(a for k in[1,2][:n]for a in range(1,3**n-8)if f(n-k)%a<1)or n

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Ridiculously slow.

Answer 8

MATL, 16 15 bytes

Oliq:",yZ\s]+]&

This solution uses 0-based indexing.

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Explanation

O        % Push the number literal 0 to the stack
l        % Push the number literal 1 to the stack
i        % Explicitly grab the input (n)
q        % Subtract 1
:        % Create the array [1...(n - 1)]
"        % For each element in this array...
  ,      % Do the following twice
    y    % Copy the stack element that is 1-deep
    Z\   % Compute the divisors
    s    % Sum the divisors
  ]      % End of do-twice loop
  +      % Add these two numbers together
]        % End of for loop
&        % Display the top stack element

Answer 9

Haskell, 64 60 bytes

d n=sum[a|a<-[1..n],mod n a<1]
f=0:scanl((.d).(+).d)1f
(!!)f

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Answer 10

PHP, 97 bytes

for($f=[0,$y=1];++$i<$argn;$x=$y,$y=$r)for($f[]=$r=$v=$n=$x+$y;--$v;)$n%$v?:$r+=$v;echo$f[$argn];

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PHP, 101 bytes

for($f=$d=[0,1];$i<$argn;$d[]=$r)for($f[]=$r=$v=$n=$d[$i]+$d[++$i];--$v;)$n%$v?:$r+=$v;echo$f[$argn];

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Answer 11

Pari/GP, 39 bytes

f(n)=if(n<3,1,sum(i=1,2,sigma(f(n-i))))

Based on Martin Ender's Mathematica answer.

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Answer 12

R, 81 bytes

f=function(n,a=1,b=1,d=numbers::divisors)`if`(n-1,f(n-1,b,sum(d(a))+sum(d(b))),a)

1-indexed, and excludes the 0 at the start of the sequence. That zero gave me a lot of trouble to implement, because the builtin numbers::divisors doesnt handle it well.

The rest is a modified version of the standard recursive function that implements the fibonacci sequence.

> f(1)
[1] 1
> f(2)
[1] 1
> f(3)
[1] 2
> f(5)
[1] 10
> f(13)
[1] 12846