Hilbert's hotel - 2020-04-05 (page 2 of 2)

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Knight

22:01

@Ante Can you prove it for special choices of $a$ and $b$ for *ANY $f(x)$ (that assumes a value of zero at least once) ?

Ante

22:13

i have found a way for general a and b but it is too tiresome to write all of that here

i can highlight the idea

hello @Peter

Knight

Please highlight it

Ante

well chose that f is linear on [a,c] with f(a)=0, and for example f(c)=-1, then choose f to be linear on [c,b] so that $\int_{a}^{b}{f(x)dx}=f(b)$

you have equation to determine c then

if that works

if not f(c) need to be chosen differently

although that could not work if f(c) is not chosen appropriately

f should be made of two linear functions

if f linear does not work then try with parabola

it should work for appropriately chosen f(c)

or, better, define f(b)=b and determine c and f(c)

you have f(a)=0 and f(b)=b and $\int_{a}^{b}{f(x)dx}=b$

so c and f(c) need to be determined

Knight

I have to understand your work, it’s tough :-)

Peter

22:33

@Ante Hi, still no result. Intuitively, I would now guess that there are large squarefree solutions.

Ante

why

Peter

the prime factors of sigma(n) and phi(n) are different from the prime factors of n, and I see nothing that prevents the sum being divisible by n.

sigma(n)/n is also unbounded

Isn't it reasonable to consider the chance for a solution to be 1/n ? Like in the case of Wieferich primes ?

Can we construct any equation not too weird with sigma(n) and phi(n) that is not already in OEIS ?

It makes more fun to analyze something new.

Ante

n*phi(n)=sigma(n)?

Peter

I noticed phi(n) | sigma(n)

but I do not know whether this is in OEIS

2phi(n) + 3sigma(n) = 6n ?

Ante

nah, (n+phi(n))|sigma(n) is better

Peter

22:44

? for(n=1,10^7,if(2*eulerphi(n)+3*sigma(n)==6*n,print1(n," ")))
14 442 1377
? for(n=1,10^7,if(Mod(sigma(n),n+eulerphi(n))==0,print1(n," ")))
2 456 828 7584 33462 1357440 1596048 1964544
?

are those in OEIS ?

Ante

the last one is not

Peter

Great ! And the first is ?

Ante

nope, also not

Peter

OK, yours is in fact more interesting.

Ante

yes

Peter

22:47

First, I will collect some solutions and then begin with conditions.

Ante

for (n+phi(n))|sigma(n) ?

Peter

yes

Solutions so far :

? for(n=1,10^8,if(Mod(sigma(n),n+eulerphi(n))==0,print(n," ",sigma(n)/(n+eulerphi(n))," ",factor(n))))
2 1 Mat([2, 1])
456 2 [2, 3; 3, 1; 19, 1]
828 2 [2, 2; 3, 2; 23, 1]
7584 2 [2, 5; 3, 1; 79, 1]
33462 2 [2, 1; 3, 2; 11, 1; 13, 2]
1357440 3 [2, 7; 3, 1; 5, 1; 7, 1; 101, 1]
1596048 2 [2, 4; 3, 1; 41, 1; 811, 1]
1964544 2 [2, 9; 3, 1; 1279, 1]

19800384 2 [2, 6; 3, 1; 281, 1; 367, 1]
26211264 2 [2, 6; 3, 1; 211, 1; 647, 1]
31451136 2 [2, 11; 3, 1; 5119, 1]

largest fraction at the moment : 3

Are all solutions even ?

complete upto 10^8

106805184 2 [2, 6; 3, 1; 167, 1; 3331, 1]

OK, good night. I will think abouth this !

Ante

bye