A256394 Prime values of pi(n) that divide n.

2, 3, 11, 67, 71, 439, 1051, 6469, 40087, 100361, 100363, 251737, 251761, 637319, 637327, 4124459, 10553513, 10553551, 27067277, 69709733, 179993171, 465769817, 3140421769, 8179002109, 8179002133, 55762149029, 55762149071, 382465573489, 1003652347081

Offset

1,1

Comments

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

Links

Giovanni Resta, Table of n, a(n) for n = 1..49

K. Gaitanas, An explicit formula for the prime counting function, arXiv:1311.1398 [math.NT], 2013.

K. Gaitanas, An Explicit Formula for the Prime Counting Function Which is Valid Infinitely Often, Amer. Math. Monthly, 122 (2015), 283.

S. W. Golomb, On the Ratio of N to pi(N), American Mathematical Monthly, 69 (1962), 36-37.

R. T. Harger and W. L. Hightower, An Interesting Property of x/pi(x), College Math. J., 40 (2009), 213-214.

Eric Weisstein's World of Mathematics, Prime Counting Function

Formula

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

Example

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

Mathematica

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}]
PrimePi[Select[Select[Range[2, 10^6], IntegerQ[#/PrimePi[#]]&], PrimeQ[PrimePi[#]]&]] (* Ivan N. Ianakiev, Apr 15 2015 *)
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 *)

Prog

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

Crossrefs

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

Sequence in context: A105217 A066046 A065597 * A219513 A001052 A184310

Adjacent sequences: A256391 A256392 A256393 * A256395 A256396 A256397

Keyword

nonn

Author

Jonathan Sondow, Apr 13 2015

Extensions

More terms from Giovanni Resta, Sep 01 2018

Status

approved