A057810 -id:A057810 - cp's OEIS frontend

A057809 Numbers k such that pi(k) divides k.

2, 4, 6, 8, 27, 30, 33, 96, 100, 120, 330, 335, 340, 350, 355, 360, 1008, 1080, 1092, 1116, 1122, 1128, 1134, 3059, 3066, 3073, 3080, 3087, 3094, 8408, 8424, 8440, 8456, 8464, 8472, 23526, 23535, 24300, 64540, 64580, 64610, 64620, 64650, 64690, 64700

Offset: 1

N. J. A. Sloane, Nov 07 2000

nonn

Each cluster of entries is approximately a power of e times larger than the previous cluster.

The sequence is infinite (Golomb, 1962). - Yifan Xie, Jun 23 2025

			120 is a member as there are exactly 30 primes less than 120 and 30 * 4 = 120.

Giovanni Resta, Table of n, a(n) for n = 1..1161 (first 296 terms from Charles R Greathouse IV)
Konstantinos N. Gaitanas, An explicit formula for the prime counting function, arXiv preprint arXiv:1311.1398 [math.NT], 2013.
S. W. Golomb, On the Ratio of N to π(N), The American Mathematical Monthly 69.1 (1962): 36-37.

Cf. A000720, A057810, A071394, A038627, A256394.

Apart from initial term same as A058011.

[n: n in [2..10^5] | n mod #PrimesUpTo(n) eq 0]; // Vincenzo Librandi, Jul 04 2016

select(t -> t mod numtheory:-pi(t) = 0, [$2..10^5]); # Robert Israel, Jul 03 2016

Select[ Range[2, 10^5], IntegerQ[ # / PrimePi[ # ]] & ]
Select[Range[1000], Divisible[#, PrimePi[#]] &] (* Requires version 6.0+. Alonso del Arte, May 24 2015 *)

is(n)=n%primepi(n)==0 \\ Charles R Greathouse IV, Sep 14 2015

More terms from James Sellers, Nov 08 2000

a(297)-a(1161) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Aug 31 2018

A065134 Remainder when n is divided by the number of primes not exceeding n.

Original entry on oeis.org

0, 1, 0, 2, 0, 3, 0, 1, 2, 1, 2, 1, 2, 3, 4, 3, 4, 3, 4, 5, 6, 5, 6, 7, 8, 0, 1, 9, 0, 9, 10, 0, 1, 2, 3, 1, 2, 3, 4, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 6, 7, 5, 6, 7, 8, 9, 10, 8, 9, 7, 8, 9, 10, 11, 12, 10, 11, 12, 13, 11, 12, 10, 11, 12, 13, 14, 15, 13, 14, 15, 16, 14, 15, 16, 17, 18, 19, 17, 18, 19

Offset: 2

Labos Elemer, Oct 15 2001

nonn

Also remainder when the number of nonprimes is divided by the number of primes (not exceeding n).

			n = 2: Pi[2] = 1,Mod[1,1] = 0, the first term = a(2) = 0; n = 100: Pi[100] = 25, Mod[100,25] = 0 = a(100); n = 20: Pi[20] = 8, Mod[20,8] = 4 = a(20).

Harry J. Smith, Table of n, a(n) for n = 2..1000

Cf. A000720, A057809, A057810, A000040, A065134, A004648, A062298, A065858-A065864, A065133.

Table[Last@ QuotientRemainder[n, PrimePi[n]], {n, 2, 91}] (* Michael De Vlieger, Jul 04 2016 *)

{ for (n=2, 1000, write("b065134.txt", n, " ", n%primepi(n)) ) } \\ Harry J. Smith, Oct 11 2009

a(n) = n (mod pi(n)).

Term a(1) removed so OFFSET changed from 1,5 to 2,4 by Harry J. Smith, Oct 11 2009

Since OFFSET is 2,4; Term a(1) removed and a(91) added by Harry J. Smith, Oct 11 2009

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

Original entry on oeis.org

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

Jonathan Sondow, Apr 13 2015

nonn

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

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

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

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

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 *)

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

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

More terms from Giovanni Resta, Sep 01 2018

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A057809 Numbers k such that pi(k) divides k.

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A065134 Remainder when n is divided by the number of primes not exceeding n.

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

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