A038606 -id:A038606 - cp's OEIS frontend

A090974 Duplicate of A038606.

0, 1, 5, 12, 31, 69, 181, 443, 1052, 2701, 6455, 15928, 40073, 100362, 251707, 637235

Offset: 0

dead

With offset 1: Index j of prime p(j) such that ceiling[p(j)/j]=n is first satisfied for n >= 2. a(n) = A038606(n) = A062742(n) = A038624(n) for n >= 3. [From Jaroslav Krizek, Dec 13 2009]

A038624 Values of pi(x) where x exceeds n * pi(x).

Original entry on oeis.org

1, 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

Jud McCranie

nonn

"Exceeds" can be interpreted as ">" or ">=" since the corresponding primes are never multiples of their indices. - R. J. Mathar, Jun 08 2008
Equivalently, a(n) = minimal k such that prime(k)/k >= n. - Enoch Haga, Oct 19 2007
a(n) = A062742(n) = A038606(n) for n >= 3. - Jaroslav Krizek, Dec 13 2009

			x exceeds 3*pi(x) when pi(x)=12, so a(3)=12

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

Cf. A038623-A038627.
Essentially the same as A062742.
Cf. A038606 (variant).

Join[{k = 1}, Table[While[Prime[k]/k < n, k++]; k, {n, 2, 18}]] (* Jayanta Basu, Jul 10 2013 *)

a(24)-a(28) from Robert G. Wilson v, Sep 26 2005
Edited by N. J. A. Sloane, Jun 30 2008 at the suggestion of R. J. Mathar.
a(29)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Sep 01 2018

A038607 a(n) is the smallest prime number k such that k > n*pi(k), where pi(k) denotes the prime counting function.

Original entry on oeis.org

2, 11, 37, 127, 347, 1087, 3109, 8419, 24317, 64553, 175211, 480881, 1304707, 3523901, 9558533, 25874843, 70115473, 189961529, 514272533, 1394193607, 3779851091, 10246935679, 27788566133, 75370121191, 204475052401, 554805820711, 1505578023841, 4086199302113, 11091501631019, 30109570413007

Offset: 1

Vasiliy Danilov (danilovv(AT)usa.net), Jul 1998

a(n) is about exp(n+1+1/(n+1)). - Charles R Greathouse IV, Sep 05 2011

			For n=1, a(1) = 2, since 2 > 1*pi(2) = 1*1. - _N. J. A. Sloane_, Dec 09 2020
For n=3, the 12th prime (37) is the first one satisfying p(k) > 3k.

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

Cf. A038606, A038623.

k = 1; Do[ While[ Prime[k] < n*k, k++ ]; Print[Prime[k]], {n, 1, 25} ]

k=1;n=1;forprime(p=3,4e9,if(p/n++>k,print1(p", ");k++)) \\ Charles R Greathouse IV, Sep 06 2011

a(n) = prime(A038606(n)) = A000040(A038606(n)).

Extended by Robert G. Wilson v and Ray Chandler, Dec 01 2004
a(26)-a(30) from Charles R Greathouse IV, Sep 05 2011, Sep 06 2011
a(31)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Sep 01 2018

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

Original entry on oeis.org

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

A090973 a(n) = ceiling(prime(n)/n).

Original entry on oeis.org

2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 1

Amarnath Murthy, Jan 04 2004

			a(12) = 4 as pi(48) = 15 > 12 > pi(36) = 11.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A038605, A038606.

Cf. A068901. - Reinhard Zumkeller, Aug 16 2009

[Ceiling(NthPrime(n)/n): n in [1..120]]; // G. C. Greubel, Feb 02 2019

Table[Ceiling[Prime[n]/n], {n, 1, 120}] (* G. C. Greubel, Feb 02 2019 *)

vector(120, n, ceil(prime(n)/n)) \\ G. C. Greubel, Feb 02 2019

[ceil(nth_prime(n)/n) for n in (1..120)] # G. C. Greubel, Feb 02 2019

For n > 1, a(n) = A038605(n)+1. - David Wasserman, Feb 23 2006

a(A038606(n)) = n+1. - Reinhard Zumkeller, Aug 16 2009

More terms from David Wasserman, Feb 23 2006

A245071 a(n) = 12n - prime(n).

Original entry on oeis.org

10, 21, 31, 41, 49, 59, 67, 77, 85, 91, 101, 107, 115, 125, 133, 139, 145, 155, 161, 169, 179, 185, 193, 199, 203, 211, 221, 229, 239, 247, 245, 253, 259, 269, 271, 281, 287, 293, 301, 307, 313, 323, 325, 335, 343, 353, 353, 353, 361, 371, 379, 385, 395, 397, 403, 409, 415, 425

Offset: 1

Freimut Marschner, Jul 21 2014

Prime(n) > n for n > 0. Let prime(n) = k*n with k as an even integer constant, for example, k = 12; then a(n) = k*n - prime(n) is a sequence of odd integers that are positive as long as k*n > prime(n). This is the case up to a(40072) = 11. If k*n < prime(n) then a(n) < 0, a(40073) = -5 up to a(40083) = -5. From a(40084) = 5 up to a(40121) = 5, a(n) > 0 again, but a(n) < 0 for n >= 40122. For k = 12 the table shows this result compared with floor(prime(n)/n) and (prime(n) mod n) <= (prime(n+1) mod (n+1)) for n >= 1. Observations:
(1) If k > floor(prime(n)/n) then a(n) is positive.
(2) If k <= floor(prime(n)/n) and (prime(n) mod n) < (prime(n+1) mod (n+1)) and n > 1 then a(n) is negative.
(3) If k <= floor(prime(n)/n) and (prime(n) mod n) > (prime(n+1) mod (n+1)) then a(n) is positive.
.
n     prime(n) floor(prime(n)/n) (prime(n) mod n)  a(n)
40072 480853        12                 5            11
40073 480881        12                23            -5
40083 481001        11             40079            -5
40084 481003        11             40074             5
40121 481447        12                 5             5
40122 481469        12                13            -5

			a(3) = 12*3 - prime(3) = 36 - 5 = 31.

Freimut Marschner, Table of n, a(n) for n = 1..100000

A000040 (prime(n)), A038605 (floor(prime(n)/n)), A004648 (prime(n) mod n), A038606 (Least k such that k-th prime > n * k), A038607 (the smallest prime number k such that k > n*pi(k)), A102281 (the largest number m such that m = pi(n*m)).

Table[12n - Prime[n], {n, 60}] (* Alonso del Arte, Jul 27 2014 *)

PARI
```
vector(133, n, 12*n-prime(n) )
```

a(n) = 12*n - prime(n).

cp's OEIS Frontend

A090974 Duplicate of A038606.

Views

Author

Keywords

Comments

A038624 Values of pi(x) where x exceeds n * pi(x).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A038607 a(n) is the smallest prime number k such that k > n*pi(k), where pi(k) denotes the prime counting function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

A090973 a(n) = ceiling(prime(n)/n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

Extensions

A245071 a(n) = 12n - prime(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula