# given a prime, finding where it is the list of primes

I'm writing a program that gives as an output the prime factorization of a number, and then I'm putting the primes into a matrix based on what prime number it is, (i.e 541 is the 100th prime, so if 541^2 divides my integer, then there would be a 2 in the 100th spot of my vector that represents my number)

Is there a function that takes as its input a prime and gives as an output where it is in the list of primes. (i.e we want f(541) = 100)

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prime_pi?
Type:           PrimePi
String form:    prime_pi
File:           /usr/local/sage-8/local/lib/python2.7/site-packages/sage/functions/prime_pi.pyx
Docstring:
The prime counting function, which counts the number of primes less
than or equal to a given value.

INPUT:

* "x" - a real number

* "prime_bound" - (default 0) a real number < 2^32, "prime_pi"
will make sure to use all the primes up to "prime_bound"
(although, possibly more) in computing "prime_pi", this can
potentially speedup the time of computation, at a cost to memory
usage.

OUTPUT:

integer -- the number of primes <= "x"


So, given your number x,an obvious "brute-force" one-liner is :

def foo(x): return map(lambda(t):prime_pi(t[0],sqrt(x)), factor(x))
bar=Integer(round(10^10*random()))
bar
7218184858
foo(bar)
[1, 1211, 31354]


HTH,

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This work! Thanks a ton!

( 2018-03-09 19:58:43 -0500 )edit

This is a solution that puts first all "possibly relevant small primes" for the task in a list, than asks for the index of a prime in the list, in a dialog with the sage interpreter:

sage: pList = prime_range(10**6)
sage: pList.index(541)
99


(So it is the pythonically $99$.th place.)

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