A038605 -id:A038605 - cp's OEIS frontend

A134914 a(n) = ceiling(n^(1/3)).

1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 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, 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, 5, 5, 5

Offset: 1

Mohammad K. Azarian, Nov 17 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A048766, A038605, A003059.

[Ceiling(n^(1/3)): n in [1..80]]; // Vincenzo Librandi, Feb 18 2013

A134914:=n->ceil(n^(1/3)): seq(A134914(n), n=1..50); # Wesley Ivan Hurt, Jul 24 2014

Table[Ceiling[n^(1/3)], {n, 80}] (* Vincenzo Librandi, Feb 18 2013 *)

a(n)=ceil(n^(1/3)) \\ Felix Fröhlich, Jul 24 2014

[ceil(n^(1/3)) for n in range(1,106)] # Stefano Spezia, Aug 17 2024

A079416 a(n) = round(prime(n)/n).

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Jan 07 2003

nonn

The sequence is not monotone, see example and A079417.

			a(20) = round(prime(20)/20) = round(71/20) = round(3.55) = 4;
a(21) = round(prime(21)/21) = round(73/21) = round(3.476190...) = 3;
a(22) = round(prime(22)/22) = round(79/22) = round(3.590909...) = 4.

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

Cf. A079418, A000040, A038605, A107609, A263076.

[Round(NthPrime(n)/n): n in [1..100]]; // G. C. Greubel, Jan 18 2019

f[n_] := Round[ Prime[n]/n]; Array[f, 105] (* Robert G. Wilson v, Oct 23 2015 *)

vector(100, n, round(prime(n)/n)) \\ G. C. Greubel, Jan 18 2019

[round(nth_prime(n)/n) for n in (1..100)] # G. C. Greubel, Jan 18 2019

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

A134917 a(n) = ceiling(n^(4/3)).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 14, 16, 19, 22, 25, 28, 31, 34, 37, 41, 44, 48, 51, 55, 58, 62, 66, 70, 74, 78, 81, 86, 90, 94, 98, 102, 106, 111, 115, 119, 124, 128, 133, 137, 142, 146, 151, 156, 161, 165, 170, 175, 180, 185, 190, 195, 200

Offset: 1

Mohammad K. Azarian, Nov 17 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A048766, A038605, A003059, A134914, A129011, A100196, A121536.

[Ceiling(n^(4/3)): n in [1..60]]; // Vincenzo Librandi, Feb 18 2013

Table[Ceiling[n^(4/3)], {n, 60}] (* Vincenzo Librandi, Feb 18 2013 *)

A134918 Ceiling(n^(5/3)).

Original entry on oeis.org

1, 4, 7, 11, 15, 20, 26, 32, 39, 47, 55, 63, 72, 82, 92, 102, 113, 124, 136, 148, 160, 173, 187, 200, 214, 229, 243, 259, 274, 290, 306, 323, 340, 357, 375, 393, 411, 430, 449, 468, 488, 508, 528, 549, 570, 591, 613, 634, 657

Offset: 1

Mohammad K. Azarian, Nov 17 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A048766, A038605, A003059, A134914, A129011, A100196, A121536, A134917.

[Ceiling(n^(5/3)): n in [1..50]]; // Vincenzo Librandi, Feb 18 2013

Table[Ceiling[n^(5/3)], {n, 50}] (* Vincenzo Librandi, Feb 18 2013 *)

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).

A024926 a(n) = Sum_{k=1..n} floor(p(k)/k).

Original entry on oeis.org

2, 3, 4, 5, 7, 9, 11, 13, 15, 17, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216

Offset: 1

Clark Kimberling

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Partial sums of A038605.

Table[Floor[Prime[n]/n],{n,70}]//Accumulate (* Harvey P. Dale, Nov 07 2019 *)

A065521 a(n) = floor(prime(n) / n) * n - prime(n) mod n.

Original entry on oeis.org

2, 1, 1, 1, 9, 11, 11, 13, 13, 11, 13, 35, 37, 41, 43, 43, 43, 47, 47, 49, 53, 53, 55, 55, 53, 55, 59, 61, 65, 67, 121, 125, 127, 133, 131, 137, 139, 141, 145, 147, 149, 155, 153, 159, 163, 169, 165, 161, 165, 171, 175, 177, 183, 181, 183, 185, 187, 193, 195, 199

Offset: 1

Reinhard Zumkeller, Nov 27 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A038605, A004648.

Table[Floor[Prime[n]/n]n-Mod[Prime[n],n],{n,60}] (* Harvey P. Dale, Dec 27 2019 *)

{ for (n=1, 1000, a=floor(prime(n) / n) * n - prime(n) % n; write("b065521.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 20 2009

a(n) = n*(prime(n)\n) - (prime(n) % n); \\ Michel Marcus, Jun 18 2018

a(n) = A038605(n) * n - A004648(n).

A067289 Numbers k such that the number of divisors of k is floor(prime(k)/k).

Original entry on oeis.org

5, 7, 11, 25, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 81, 207, 212, 236, 242, 243, 244, 245, 261, 268, 275, 279, 284, 292, 316, 325, 332, 333, 338, 356, 363, 369, 387, 388, 404, 412, 423, 425, 428, 436, 729, 1065, 1066, 1070, 1074, 1085, 1086, 1090

Offset: 1

Benoit Cloitre, Feb 24 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A038605.

Select[Range[1000], DivisorSigma[0, #] == Floor[Prime[#]/#] &] (* Amiram Eldar, Apr 23 2022 *)

Numbers k such that A000005(k) = floor(prime(k)/k).

A076080 a(n) = A076079(n)/n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 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, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 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

Offset: 1

Amarnath Murthy, Oct 05 2002

nonn

Cf. A076079.

Except for first term, same as A038605.

1,seq(floor(evalf(ithprime(n)/n,100)),n=2..200);

a(n) = floor((prime(n)-1)/n). - David Wasserman, Feb 23 2006

More terms from Sascha Kurz, Jan 30 2003

More terms from David Wasserman, Feb 23 2006

cp's OEIS Frontend

A134914 a(n) = ceiling(n^(1/3)).

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A079416 a(n) = round(prime(n)/n).

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

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A134917 a(n) = ceiling(n^(4/3)).

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Magma

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A134918 Ceiling(n^(5/3)).

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Magma

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

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A024926 a(n) = Sum_{k=1..n} floor(p(k)/k).

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A065521 a(n) = floor(prime(n) / n) * n - prime(n) mod n.

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