A038625 -id:A038625 - cp's OEIS frontend

A062742 Index j of prime p(j) such that floor(p(j)/j) = n is first satisfied.

2, 1, 12, 31, 69, 181, 443, 1052, 2701, 6455, 15928, 40073, 100362, 251707, 637235, 1617175, 4124437, 10553415, 27066974, 69709680, 179992909, 465769803, 1208198526, 3140421716, 8179002096, 21338685407, 55762149030, 145935689361

Offset: 1

Labos Elemer, Jul 12 2001

nonn

			The q(j)=p(j)/j quotient when the value 14 first appears: {j=251706, p(j)=3523841, q(j)=13.9998291} {251707, 3523901, 14.0000119} {251708, 3523903, 13.9999642} {251709, 3523921, 13.9999801} {251710, 3523957, 14.0000675} {251711, 3523963, 14.0000357}

Giovanni Resta, Table of n, a(n) for n = 1..50

Essentially the same as A038624.

Cf. A038606. - R. J. Mathar, Jan 30 2009

{a062742(m)=local(n,j); for(n=1,m,j=1; while(floor(prime(j)/j)!=n,j++); print1(j,","))} a062742(10^7)

a(n) = Min_{j| floor(p(j)/j) = n}. Note that neither p(j)/j nor floor(p(j)/j) is monotonic.
a(n) = pi(A062743(n)).
a(n) = A038606(n) = A038624(n) for n >= 3. - Jaroslav Krizek, Dec 13 2009

More terms from Jason Earls, May 15 2002
a(17)-a(28) from Farideh Firoozbakht and Robert G. Wilson v, Sep 13 2005
a(29)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Sep 01 2018

A062743 Smallest prime prime(m) such that floor(prime(m)/m) = n.

Original entry on oeis.org

3, 2, 37, 127, 347, 1087, 3109, 8419, 24317, 64553, 175211, 480881, 1304707, 3523901, 9558533, 25874843, 70115473, 189961529, 514272533, 1394193607, 3779851091, 10246935679, 27788566133, 75370121191, 204475052401

Offset: 1

Labos Elemer, Jul 12 2001

nonn

a(n+1)/a(n) -> e as n -> infinity, as do the m's.

Giovanni Resta, Table of n, a(n) for n = 1..50 (first 29 terms from Robert G. Wilson v)

Essentially the same as A038623.

Cf. A038607, A038625.

Do[ k = 1; While[ Floor[ Prime[m]/ m] != n, m++ ]; Print[Prime[k] ], {n, 1, 27} ]

A062742(n) = pi(a(n)).

More terms from Robert G. Wilson v, Jul 13 2001
a(27) from Farideh Firoozbakht, Sep 12 2005
Corrected by T. D. Noe, Nov 14 2006
a(30)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Sep 01 2018

A073436 Smallest k such that k mod pi(k) = n.

Original entry on oeis.org

2, 3, 5, 7, 16, 21, 22, 25, 26, 29, 32, 65, 66, 70, 77, 78, 82, 86, 87, 88, 92, 93, 94, 95, 99, 106, 116, 117, 118, 119, 218, 219, 220, 221, 222, 247, 248, 249, 250, 255, 256, 261, 262, 267, 268, 289, 290, 291, 292, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306

Offset: 0

Labos Elemer, Jul 31 2002

nonn

a(n) > a(n-1) except for 68, 180, 1051, 6454, 6456, 6459, 40073, 40078, ..., . - Robert G. Wilson v, Feb 24 2023

			Remainder 7 appears first as 25 mod pi(25) = 25 mod 9 = 7, so a(7) = 25.

Giovanni Resta, Table of n, a(n) for n = 0..10000

Cf. A000720, A038625, A057809, A065134, A073435, A073437, A360789.

with(numtheory); f:=proc(n) local i,j,k; for i from 2 to 10000 do if i mod pi(i) = n then RETURN(i); fi; od: RETURN(-1); end; # N. J. A. Sloane, Sep 01 2008

a = Compile[{{n, Integer}}, Block[{k = 2}, While[ Mod[k, PrimePi@ k] != n, k++]; k]]; Array[a, 59, 0] (* _Robert G. Wilson v, Feb 24 2023 *)

a(n)={my(q=0, k=2);forprime(p=3, oo, q++; while(kAndrew Howroyd, Feb 23 2023

a(n) = Min{k: k mod A000720(k) = n} = Min{k: A065134(k) = n}.

a(0) from Robert G. Wilson v, Feb 23 2023

A360789 Least prime p such that p mod primepi(p) = n.

Original entry on oeis.org

2, 3, 5, 7, 379, 23, 401, 61, 59, 29, 67, 71, 467, 79, 83, 179, 431, 89, 176557, 191, 24419, 491, 97, 101, 499, 1213, 3169, 3191, 523, 229, 3187, 223, 3203, 8609, 3163, 251, 176509, 257, 24509, 263, 3253, 269, 547, 3347, 1304867, 293

Offset: 0

Robert G. Wilson v, Feb 20 2023

nonn

Inspired by A048891.

			For n=0, prime p=2 has p mod primepi(p) = 2 mod 1 = 0 so that a(0) = 2.
For n=4, no prime has p mod primepi(p) = 4 until reaching p=379 which is 379 mod 75 = 4, so that a(4) = 379.

Robert G. Wilson v, Table of n, a(n) for n = 0..19217

Cf. A038625, A004648, A048891, A052013, A073325, A073436, A162567, A171430, A171431, A171432, A171434.

V:= Array(0..100): count:= 0:
p:= 1:
for k from 1 while count < 101 do
  p:= nextprime(p);
  v:= p mod k;
  if v <= 100 and V[v] = 0 then V[v]:= p; count:= count+1 fi;
od:
convert(V,list); # Robert Israel, Feb 28 2023

t[_] := 0; p = 2; pi = 1; While[p < 1400000, m = Mod[p, pi]; If[m < 100 && t[m] == 0, t[m] = p]; p = NextPrime@p; pi++]; t /@ Range[0, 99]

a(n)={my(k=n); forprime(p=prime(n+1), oo, k++; if(p%k ==n, return(p)))} \\ Andrew Howroyd, Feb 21 2023

from sympy import prime, nextprime
def A360789(n):
    p, m = prime(n+1), n+1
    while p%m != n:
        p = nextprime(p)
        m += 1
    return p # Chai Wah Wu, Mar 18 2023

a(n) = prime(A073325(n+1)). - Kevin Ryde, Feb 21 2023

A086511 a(n) is the smallest integer k > 1 such that k > n * pi(k), where pi() denotes the prime counting function.

Original entry on oeis.org

2, 9, 28, 121, 336, 1081, 3060, 8409, 23527, 64541, 175198, 480865, 1304499, 3523885, 9557956, 25874753, 70115413, 189961183, 514272412, 1394193581, 3779849620, 10246935645, 27788566030, 75370121161, 204475052376, 554805820453, 1505578023622, 4086199301997

Offset: 1

Tim Paulden (timmy(AT)cantab.net), Sep 09 2003

nonn

a(n) is bounded above by the sequence A038623, in which k is required to be prime. In addition, the sequence pi(a(n)) = {1, 4, 9, 30, 67, 180, 437, 1051, ...} closely resembles the sequence A038624, in which the n-th term is the minimal t such that k >= n * pi(k) for every k satisfying pi(k) = t. If we were to make the inequality in A038624 strict, the resulting sequence would provide an upper bound for pi(a(n)). Sequences A038625, A038626 and A038627 focus on the equality k = n * pi(k): as we would expect, a(n) follows A038625 very closely for large n.

			Consider the pairs (k, pi(k)) for k > 1. The inequality k > 1 * pi(k) is first satisfied at k = 2 and so a(1) = 2. Similarly, the inequality k > 2 * pi(k) is first satisfied at k = 9 and so a(2) = 9.

Giovanni Resta, Table of n, a(n) for n = 1..50
Eric Weisstein's World of Mathematics, Prime Counting Function.

Cf. A038623, A038624, A038625, A038626, A038627.

a(n) = { k = 2; while (k <= n*primepi(k), k++); return (k);} \\ Michel Marcus, Jun 19 2013

Heuristically, for large n, a(n) ~= 3.0787*(2.70888^n) [error < 0.05% for 15 <= n <= 20].
From Nathaniel Johnston, Apr 10 2011: (Start)
a(n) >= exp(n/2 + sqrt(n^2 + 4n)/2), n >= 6.
a(n) = A038625(n) + m(n)*n + 1 for some m(n) >= 0. For n = 2, 3, 4, ..., m(n) = 3, 0, 6, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, ...
(End)

a(21)-a(26) from Nathaniel Johnston, Apr 10 2011
Corrected a(26) and a(27)-a(28) from Giovanni Resta, Sep 01 2018
a(29)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Sep 01 2018

A087236 a(n) is the difference between the largest and smallest integer solutions to n=x/pi(x), where pi(x) = A000720(x).

Original entry on oeis.org

6, 6, 24, 30, 126, 35, 64, 774, 180, 0, 600, 221, 770, 2145, 32, 4573, 8172, 5852, 5720, 7035, 792, 7774, 5256, 2825, 104, 2484, 1008, 2088, 8880, 9176, 10464, 759, 68, 5880, 23688, 28490, 3420, 49686, 58160, 62074, 136878, 26316, 264, 130320, 16882, 705, 96528, 14063, 95750

Offset: 2

Labos Elemer, Sep 04 2003

nonn

			n=22: a(22) = 10246936436-10246935644 = 792 = 22*36.
a(2) = 6 since x/pi(x) = 2 for x = {2,4,6,8}; 8 - 2 = 6. - _Michael De Vlieger_, Mar 25 2017

Cf. A038623-A038627, A057809, A087235-A087240.

Last@ # - First@ # & /@ Values@ Rest@ KeySort@ PositionIndex@ Table[n/PrimePi[n] /. k_ /; Not@ IntegerQ@ k -> 0, {n, 2, 10^6}] (* Michael De Vlieger, Mar 25 2017, Version 10 *)

a(n) = Max{x; n*pi(n)=x} - Min{x; n*pi(n)=x} = A038625(n) - A087235(n).

a(n) is divisible by n, the quotients are in A087237.

a(27)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Sep 01 2018

A334598 a(n) is the largest nonnegative integer m such that m >= pi(m)^(1 + 1/n).

Original entry on oeis.org

4, 28, 1860, 149052, 12771496, 1221908916, 132662942122, 16354869261256, 2272946910544740, 353076161059625536, 60799066209732571716, 11518836088596729968092

Offset: 1

Eduard Roure Perdices, May 07 2020

For a nonnegative integer m, pi(m) = A000720(m). It is well-known that if
  m >= 17, then m/log(m) < pi(m). [Rosser and Schoenfeld]
Fix a real exponent d > 0. If m is big enough, then m < (m/log(m))^(1 + d). In particular, choosing d = 1/n, with n >= 1, we deduce that a(n) exists.
Note that different choices of the exponent d will produce analogous sequences.
The estimates of pi(m) in [Dusart, Thm. 5.1] and [Axler, Thm. 2] allow us to obtain upper and lower bounds for a(n). In particular, we can conclude that in base 10:
  a(13) has 25 digits, starting with 238;
  a(14) has 27 digits, starting with 536;
  a(15) has 30 digits, starting with 1304;
  a(16) has 32 digits, starting with 3409.
The tool primecount [Walisch], used to compute pi(10^28) in A006880, can handle pi(m) for m <= 10^31, and since (a(n)) is monotonically increasing, it seems that the computation of a(n) for n >= 16 will be challenging.
It is easy to see that for every n >= 1, a(n) is even and a(n)+1 is prime. - Eduard Roure Perdices, Nov 07 2021

Christian Axler, Estimates for pi(x) for large values of x and Ramanujan's prime counting inequality, Integers 18 (2018), Paper No. A61, 14 pp.
Pierre Dusart, Explicit estimates of some functions over primes, The Ramanujan Journal 45 (2018), no. 1, 227-251.
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), no. 1, 64-94.
Kim Walisch, primecount, Github, Aug 14 2021.

Cf. A334599, A087235, A038625.

a(8) from Giovanni Resta, May 07 2020
a(9)-a(10) from Daniel Suteu, May 20 2020
a(11)-a(12) from Eduard Roure Perdices, Nov 07 2021

A334599 a(n) is the largest nonnegative integer m such that m - pi(m) >= pi(m)^(1 + 1/n).

Original entry on oeis.org

2, 2, 346, 66942, 7087878, 744600720, 85281842598, 10892966758462, 1553240096780862, 246080334487930558, 43047454015229292840, 8262178422446205100776

Offset: 1

Eduard Roure Perdices, May 07 2020

For a nonnegative integer m, pi(m) = A000720(m). It is well-known that if
  m >= 17, then m/log(m) < pi(m). [Rosser and Schoenfeld]
Fix a real exponent d > 0. If m is big enough, then m < (m/log(m)) + (m/log(m))^(1 + d). In particular, choosing d = 1/n, with n >= 1, we deduce that a(n) exists.
Note that different choices of the exponent d will produce analogous sequences.
The estimates of pi(m) in [Dusart, Thm. 5.1] and [Axler, Thm. 2] allow us to obtain upper and lower bounds for a(n). In particular, we can conclude that in base 10:
  a(13) has 25 digits, starting with 1729;
  a(14) has 27 digits, starting with 392;
  a(15) has 29 digits, starting with 962;
  a(16) has 32 digits, starting with 2534.
The tool primecount [Walisch], used to compute pi(10^28) in A006880, can handle pi(m) for m <= 10^31, and since (a(n)) is monotonically increasing, it seems that the computation of a(n) for n >= 16 will be challenging.
It is easy to see that for every n >= 1, a(n) is even and a(n)+1 is prime. - Eduard Roure Perdices, Nov 07 2021

Christian Axler, Estimates for pi(x) for large values of x and Ramanujan's prime counting inequality, Integers 18 (2018), Paper No. A61, 14 pp.
Pierre Dusart, Explicit estimates of some functions over primes, The Ramanujan Journal 45 (2018), no. 1, 227-251.
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), no. 1, 64-94.
Kim Walisch, primecount, Github, Aug 14 2021.

Cf. A334598, A087235, A038625.

a(8) from Giovanni Resta, May 07 2020
a(9)-a(10) from Daniel Suteu, May 20 2020
a(11)-a(12) from Eduard Roure Perdices, Nov 07 2021

A121971 a(n) is the smallest number such that pi(n) divides n a record-breaking prime number of times.

Original entry on oeis.org

2, 27, 330, 3059, 175197, 1304498, 70115412, 514272411, 27788566029, 11091501630949, 81744303089590, 32781729631790293, 1784546064357412171, 13169525310647352914, 717466145742128063267

Offset: 1

G. L. Honaker, Jr., Sep 04 2006

			a(3)=330 because pi(330)=66 and 330 divided by 66 is 5, the first prime greater than the prime quotient in a(2) which was 3.

Cf. A000720, A000040, A038625.

a(n) = A038625(A000040(n)). - Ridouane Oudra, Aug 11 2019

a(10)-a(11) from Donovan Johnson, Dec 02 2009

a(12)-a(15) from Ridouane Oudra, Aug 11 2019

A176614 Conjectured least number M such that PrimePi(m)*log(m)/m lies between 1+(1/n) and 1-(1/n) for all m >= M.

Original entry on oeis.org

1, 3, 3, 114, 294, 1674, 5904, 18596, 60181, 159875, 452639, 1297696, 3515851, 9840788, 27100113, 74045962, 198875487, 541400081

Offset: 1

Michael B. Porter, Apr 21 2010

The Prime Number Theorem guarantees that the terms of this sequence exist. It states that the limit of PrimePi(x)*log(x)/x as x goes to infinity exists and is equal to 1. The existence of M follows from the definition of a limit. The terms a(n) listed are taken from a table of PrimePi(m)*log(m)/m up to m=2*10^6.
The function PrimePi(m)*log(m)/m is less than 1.25506 for all m > 1 and is greater than 1 for all m >= 17, so the first three a(n) are proved (see the Weisstein reference).
Using the bounds given by Dusart (2010), each of the terms a(1)-a(18) has been verified to be the minimal number M for n = 1..18. - Chai Wah Wu, Apr 27 2018

Pierre Dusart, Estimates of Some Functions Over Primes without R.H., arXiv:1002.0442 [math.NT], 2010.
Eric W. Weisstein, Prime Counting Function

Cf. A000720, A038625, A006880.

/* generates a(12) */
nmax=0;
for(n=1,2000000,v=primepi(n)*log(n)/n;if(v<1-(1/12)||v>1+(1/12),nmax=n));
print(nmax+1)

a(13)-a(18) from Chai Wah Wu, Apr 27 2018

cp's OEIS Frontend

A062742 Index j of prime p(j) such that floor(p(j)/j) = n is first satisfied.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A062743 Smallest prime prime(m) such that floor(prime(m)/m) = n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A073436 Smallest k such that k mod pi(k) = n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A360789 Least prime p such that p mod primepi(p) = n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A086511 a(n) is the smallest integer k > 1 such that k > n * pi(k), where pi() denotes the prime counting function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A087236 a(n) is the difference between the largest and smallest integer solutions to n=x/pi(x), where pi(x) = A000720(x).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A334598 a(n) is the largest nonnegative integer m such that m >= pi(m)^(1 + 1/n).

Views

Author

Keywords

Comments

Links

Crossrefs