euler.txt

Euler Problems

Problem 1
Find the sum of all multiples of 3 or 5 below 1000.
{+/ ((0=3|⍳⍵)∨0=5|⍳⍵) / ⍳⍵} 999
233168

Problem 2
By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued terms.
+/0=2|((fib1⍣33)1)/((fib1⍣33)1)
4613732

where,
  fib1 ← {⍵,+/¯2↑⍵}
  fib2 ← {⊃(+/,⊃)⍣⍵⊢1}

{(fib2 ⍵) >4e6: ⍵ ⋄ ∇⍵+1}1
33

(fib1⍣33)1
1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040 1346269 2178309 3524578 5702887

Problem 3
The prime factors of 13195 are 5, 7, 13 and 29.

What is the largest prime factor of the number 600851475143 ?

Using Dyalog function
factors←{⎕ML ⎕IO←1          ⍝ Prime factors of ⍵.
    ⍵{                      ⍝ note: ⎕wa>(⍵*÷2)×2*4.
        ⍵,(⍺÷×/⍵)~1         ⍝ append factor > sqrt(⍵).
    }∊⍵{                    ⍝ concatenated,
        (0=(⍵*⍳⌊⍵⍟⍺)|⍺)/⍵   ⍝ powers of each prime factor.
    }¨⍬{                    ⍝ remove multiples:
        nxt←⊃⍵              ⍝ next prime, and
        msk←0≠nxt|⍵         ⍝ ... mask of non-multiples.
        ∧/1↓msk:⍺,⍵         ⍝ all non multiples - finished.
        (⍺,nxt)∇ msk/⍵      ⍝ sieve remainder.
    }⍵{                     ⍝ from,
        (0=⍵|⍺)/⍵           ⍝ divisors of ⍵ in:
    }2,(1+2×⍳⌊0.5×⍵*÷2),⍵   ⍝ 2,3 5 .. sqrt(⍵),⍵
}
71 839 1471 6857