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Prime values of pi(n) that divide n.
4

%I #28 Feb 21 2023 02:10:21

%S 2,3,11,67,71,439,1051,6469,40087,100361,100363,251737,251761,637319,

%T 637327,4124459,10553513,10553551,27067277,69709733,179993171,

%U 465769817,3140421769,8179002109,8179002133,55762149029,55762149071,382465573489,1003652347081

%N Prime values of pi(n) that divide n.

%C a(n) is the largest prime factor of n, since pi(n) ~ n / log n.

%H Giovanni Resta, <a href="/A256394/b256394.txt">Table of n, a(n) for n = 1..49</a>

%H K. Gaitanas, <a href="http://arxiv.org/abs/1311.1398">An explicit formula for the prime counting function</a>, arXiv:1311.1398 [math.NT], 2013.

%H K. Gaitanas, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.122.03.283">An Explicit Formula for the Prime Counting Function Which is Valid Infinitely Often</a>, Amer. Math. Monthly, 122 (2015), 283.

%H S. W. Golomb, <a href="http://www.jstor.org/stable/2312732">On the Ratio of N to pi(N)</a>, American Mathematical Monthly, 69 (1962), 36-37.

%H R. T. Harger and W. L. Hightower, <a href="http://www.jstor.org/stable/25653721">An Interesting Property of x/pi(x)</a>, College Math. J., 40 (2009), 213-214.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeCountingFunction.html">Prime Counting Function</a>

%F a(n) = A000720(A071394(n)) = A006530(A071394(n)).

%e pi(6) = 3 is prime, and 3 divides 6, so 3 is a member.

%t c = 0; lpf[n_] := If[ PrimeQ[n], c++; n, Transpose[ FactorInteger[n]][[1, -1]]]; Do[ If[lpf[n] == c, Print[ PrimePi[n]]], {n, 2, 10^7}]

%t PrimePi[Select[Select[Range[2,10^6],IntegerQ[#/PrimePi[#]]&],PrimeQ[PrimePi[#]]&]] (* _Ivan N. Ianakiev_, Apr 15 2015 *)

%t Select[Table[{PrimePi[n],n},{n,10^6}],PrimeQ[#[[1]]]&&Divisible[#[[2]],#[[1]]]&][[All,1]] (* The program generates the first 9 terms of the sequence. To generate more, increase the constant for n. *) (* _Harvey P. Dale_, Feb 08 2022 *)

%o (PARI) for(n=1,10^6,if(isprime(p=primepi(n))&&!(n%primepi(n)),print1(p,", "))) \\ _Derek Orr_, Apr 14 2015

%Y Cf. A000720, A006530, A038623-A038627, A057809, A057810, A071394, A087235-A087241.

%K nonn

%O 1,1

%A _Jonathan Sondow_, Apr 13 2015

%E More terms from _Giovanni Resta_, Sep 01 2018