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Happy CPFC day! (common number of lifetime's prime-factors)

Posted 3 years ago

Have some weekend fun figuring out when you share a day in life allowing a factorization that has an equal number of prime factors as other people around you. Take on the challenge to find the asymptotic probability that a day with common given prime factor count will appear in the future given the differences between birthdays and the calendar date you are interested in. Best regards, Fabian

POSTED BY: Fabian Wenger
5 Replies

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POSTED BY: Moderation Team
Posted 3 years ago

Easier to read.

$\large\int_2^{\sqrt[3]{M}} -\frac{\log \left(\frac{\log (p)}{\log \left(\frac{M}{p^2}\right)}\right)}{p \log (p) \log \left(\frac{M}{p}\right)} \, dp$

POSTED BY: Rohit Namjoshi

Hint: The probability to randomly hit a day with a prime factor count of 2 if the number chosen is M is about Log[Log[M]/Log[2]-1]/Log[M]. There are more exact asymptotic forms but it may get rather messy. Here a numerical comparison for this approximate probability with a numerical one when averaging over the closest 10^4 integers as a function of M: 2 prime factor count probability asymptotics

For a 3-prime factor count I cannot get out a closed form expression but an integral that can be evaluated numerically:

\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(2\), 
SuperscriptBox[\(M\), \(1/3\)]]\(\(-
\*FractionBox[\(Log[
\*FractionBox[\(Log[p]\), \(Log[
\*FractionBox[\(M\), 
SuperscriptBox[\(p\), \(2\)]]]\)]]\), \(p\ Log[
\*FractionBox[\(M\), \(p\)]]\ Log[p]\)]\) \[DifferentialD]p\)\)

and the numerical comparison is given in the following figure:

3 prime factor count asymptotics

POSTED BY: Fabian Wenger
Posted 3 years ago

Hi Fabian,

Cool idea! I have been trying it out on extended family members birthdates.

A more functional way to generate addDays.

addDays = 
 Subsets[bDates, {2}] // Map[-DateDifference[Sequence @@ ##] & /* QuantityMagnitude]
POSTED BY: Rohit Namjoshi

Thanks Rohit, always interesting to learn how to write better WL code, both Subsets and Sequence were new for me!

POSTED BY: Fabian Wenger
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