Herhangi bir n pozitif tamsayısı için, pozitif bölenlerinin kareleri toplamı σ2(n) ile gösterilsin. Örneğin,
Cevap: 1922364685
C++:
σ2(10) = 1 + 4 + 25 + 100 = 130.
σ2(n) tamkare olmak üzere, 0 < n < 64,000,000 için tüm n sayılarının toplamını bulunuz.Cevap: 1922364685
C++:
#include <stdio.h>
#include <math.h>
int main() {
int isprime[8096],primes[1028],A[8000],i,j,q,n,up,np;
long long int S[8000],sum=0,mul,div,p,sq;
for(i=0;i<8096;i++) isprime[i]=1;
isprime[0]=0;
isprime[1]=0;
np=0;
for(i=2;i<8096;i++) {
if(isprime[i]) {
for(j=i*i;j<8096;j+=i) isprime[j]=0;
primes[np]=i;
np++;
}
}
for(n=0;n<64000000;n+=8000) {
for(i=0;i<8000;i++) A[i]=1;
for(i=0;i<8000;i++) S[i]=1;
up=n+8000;
for(i=0;primes[i]*primes[i]<=up;i++) {
p=primes[i];
q=1;
mul=1;
while(q<=up/p) {
q*=p;
div=mul;
mul=mul*p*p+1;
for(j=((n+q-1)/q)*q-n;j<8000;j+=q) S[j]/=div,S[j]*=mul,A[j]*=p;
}
}
for(i=(n==0);i<8000;i++) {
if(A[i]!=n+i) {
p=(n+i)/A[i];
S[i]*=p*p+1;
}
sq=(long long int) sqrt(S[i]);
if(sq*sq==S[i]) sum+=n+i;
}
}
printf("%I64d\n",sum);
return 0;
} Mathematica:sqrs = 0;
For[x = 1, x < 64000000, x++, {
tot = DivisorSigma[2, x];
If[Sqrt[tot] == IntegerPart[Sqrt[tot]], sqrs += x];
}];
Print[sqrs] Pascal:program bidon;
uses minimaple;
var t:array[1..64000000] of int64;
function longueur(n:int64):int64;
var som:int64;
begin
som:=0;
while n<>0 do
begin
som:=som+1;
n:=n div 10;
end;
longueur:=som;
end;
function heron(n:int64):int64;
var x:extended;
taille,mem,k:int64;
begin
taille:=longueur(n);
x:=exporapide(10,taille);
mem:=trunc(x);
x:=(x+n/x)/2;
k:=trunc(x);
while mem<>k do
begin
mem:=trunc(x);
x:=(x+n/x)/2;
k:=trunc(x);
end;
heron:=abs(k);
end;
function issqr(n:int64):boolean;
var k:int64;
begin
k:=heron(n);
if k*k=n then issqr:=true else issqr:=false;
end;
function cb(n:int64;p:int64):int64;
var som:int64;
begin
som:=0;
while n mod p=0 do
begin
som:=som+1;
n:=n div p;
end;
cb:=som;
end;
function rez(n:int64):int64;
var i,som,j,k,ad,u:int64;
flag:boolean;
begin
som:=1;
t[1]:=1;
i:=2;
while i<=n do
begin
if i mod 100000=0 then writeln(i);
if isprime(i)=true then t[i]:=i*i+1
else
begin
j:=1;
flag:=false;
while flag=false do
begin
j:=nextprime(j);
if i mod j=0 then flag:=true;
end;
k:=cb(i,j);
ad:=0;
u:=0;
while u<=k do
begin
ad:=ad+exporapide(j,2*u);
u:=u+1;
end;
k:=t[i div exporapide(j,k)]*ad;
if issqr(k) then som:=som+i;
t[i]:=k;
end;
i:=i+1;
end;
rez:=som;
end;
begin
writeln(rez(64000000));
end. Java:uses getime;
const n=64000000;
var a:array[1..n] of QWord;
i,j:longint;
r,i2,ii:QWord;
function issquare(n:QWord):boolean;
var r:QWord;
begin
r:=trunc(sqrt(n));
if r*r=n then issquare:=true else issquare:=false;
end;
begin
for i:=1 to n do
begin
ii:=i;
i2:=ii*ii;
for j:=1 to n DIV i do
a[i*j]:=a[i*j]+i2;
end;
r:=1;
for i:=2 to n do
if issquare(a[i]) then r:=r+i;
writeln(r);
end. Matlab:clc
clear all
tic
limit=64000000;
a=ones(limit,1);
starting=1:limit;
for flag=2:limit
increment=flag:flag:limit;
a(increment)=a(increment)+flag.^2;
end
b=sqrt(a);
final_answer=sum(starting(abs(b-round(b))<1e-9))
toc Basic:Private Function GetValue() As Long
Dim sup As Long = 64000000
Dim lstPrimes As List(Of Long) = FewFirstPrimes(CLng(Sqrt(sup)))
Dim lstUsedPrimes As New List(Of Long)
Dim lstLeftPrimes As New List(Of Long)
lstLeftPrimes.AddRange(lstPrimes)
Dim counter As Long = 1 'that's because 1 is also a square
For Each p In lstPrimes
For k As Long = 1 To DetermineUpperPowerLimit(p, sup)
Dim q As Long = 0
Dim isOver As Boolean = False
Do Until isOver
q += 1
Dim wasUsed As Boolean = False
For Each up In lstUsedPrimes
If q Mod up = 0 Then
wasUsed = True
Exit For
End If
Next
If (q Mod p <> 0) AndAlso Not wasUsed Then
Dim d As Long = CType((q * Pow(p, k)), Long)
If d >= sup Then
isOver = True
Else
Dim sum As Long = GetSQforPrimePower(p, k)
'-------------------------------------------
Dim m As Long = q
For Each pInner In lstLeftPrimes
If m Mod pInner = 0 Then
Dim isOverInner As Boolean = False
Dim temp As Long = CLng(m / pInner)
Dim kInner As Long = 1
Do Until isOverInner
If temp Mod pInner <> 0 Then
isOverInner = True
Else
kInner += 1
temp = CLng(temp / pInner)
End If
Loop
sum *= GetSQforPrimePower(pInner, kInner)
m = CLng(m / Pow(pInner, kInner))
End If
If m = 1 OrElse pInner > Sqrt(m) Then
Exit For
End If
Next
'if m is not 1 then it is a prime
If m > 1 Then
sum *= GetSQforPrimePower(m, 1)
End If
'------------------------------------------------------
Dim result As Double = Math.Sqrt(sum)
If result Mod 1 = 0 Then
counter += d
End If
End If
End If
Loop
Next
lstUsedPrimes.Add(p)
lstLeftPrimes.Remove(p)
Next
Return counter
End Function
Private Function DetermineUpperPowerLimit(ByVal d As Long, ByVal sup As Long) As Long
Dim p As Long = 0
Dim isOver As Boolean = False
Do Until isOver
Dim temp As Long = CLng(Math.Pow(d, p))
If temp >= sup Then
isOver = True
Else
p = p + 1
End If
Loop
Return p - 1
End Function
Private Function GetSQforPrimePower(ByVal p As Long, ByVal k As Long) As Long
Return CLng((Math.Pow(p, (k + 1) * 2) - 1) / (Math.Pow(p, 2) - 1))
End Function
Public Function IsPrime(ByVal num As Long) As Boolean
If num = 2 Or num = 3 Then
Return True
End If
If ((num And 1) = 0) Or ((num Mod 3) = 0) Or (num < 2) Then
Return False
End If
For i As Long = 6 To CInt(Math.Ceiling(Math.Sqrt(num)) + 1) Step 6
If (num Mod (i - 1)) = 0 Then
Return False
End If
If (num Mod (i + 1)) = 0 Then
Return False
End If
Next
Return True
End Function
Public Function FewFirstPrimes(ByVal Supremum As Long) As List(Of Long)
Dim loPrimes As New List(Of Long)
For i As Long = 1 To Supremum
If IsPrime(i) Then
loPrimes.Add(i)
End If
Next
Return loPrimes
End Function