

A190413


primepi(R_{n*m}) <= n*primepi(R_m) for m >= a(n), where R_k is the kth Ramanujan prime (A104272).


2



1, 1245, 189, 189, 85, 85, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
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OFFSET

1,2


COMMENTS

This is Conjecture 1 in the paper by Sondow, Nicholson, and Noe. The conjecture has been verified for n <= 20 and Ramanujan primes less than 10^9.
A restatement is rho(n*m) <= n*rho(m) for m >= a(n), where rho = A179196.
The conjecture has been proven for n > 10^300 by Shichun Yang and Alain Togbé.  Jonathan Sondow, Jan 21 2016
The conjecture has been proven for n > 38 and m > 9 by Christian Axler. Complete exception list can be found in remark of paper.  John W. Nicholson, Aug 04 2019


LINKS



FORMULA

For all n >= 20, a(n) = 2.


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



