A050502 -id:A050502 - cp's OEIS frontend

A212455 a(n) = floor(2n*log(2n)).

1, 5, 10, 16, 23, 29, 36, 44, 52, 59, 68, 76, 84, 93, 102, 110, 119, 129, 138, 147, 156, 166, 176, 185, 195, 205, 215, 225, 235, 245, 255, 266, 276, 286, 297, 307, 318, 329, 339, 350, 361, 372, 383, 394, 404, 416, 427

Offset: 1

Mohammad K. Azarian, May 31 2012

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

Cf. A050502, A050504, A212456, A212457, A212458, A212459, A212460, A212461, A212462.

PROG(y := [], x := 100, LOOP(IF(x = 0, RETURN y), y := ADJOIN(FLOOR(2·x·LOG(2·x)), y), x := x - 1))

[Floor(2*n*Log(2*n)): n in [1..80]]; // Vincenzo Librandi, Feb 13 2013

Table[Floor[2*n*Log[2*n]], {n, 60}] (* Vincenzo Librandi, Feb 13 2013 *)

a(n)=2*n*log(2*n)\1 \\ Charles R Greathouse IV, Sep 04 2015

A212456 a(n) = floor(3n*log(3n)).

Original entry on oeis.org

3, 10, 19, 29, 40, 52, 63, 76, 88, 102, 115, 129, 142, 156, 171, 185, 200, 215, 230, 245, 261, 276, 292, 307, 323, 339, 355, 372, 388, 404, 421, 438, 454, 471, 488, 505, 522, 539, 557, 574, 591, 609, 626, 644, 662, 679

Offset: 1

Mohammad K. Azarian, May 31 2012

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

Cf. A050502, A050504, A212455, A212457, A212458, A212459, A212460, A212461, A212462.

PROG(y := [], x := 100, LOOP(IF(x = 0, RETURN y), y := ADJOIN(FLOOR(3·x·LOG(3·x)), y), x := x - 1))

[Floor(3*n*Log(3*n)): n in [1..80]]; // Vincenzo Librandi, Feb 13 2013

Table[Floor[3*n*Log[3*n]], {n, 60}] (* Vincenzo Librandi, Feb 13 2013 *)

a(n)=3*n*log(3*n)\1 \\ Charles R Greathouse IV, Sep 04 2015

A212457 a(n) = floor(4n*log(4n)).

Original entry on oeis.org

5, 16, 29, 44, 59, 76, 93, 110, 129, 147, 166, 185, 205, 225, 245, 266, 286, 307, 329, 350, 372, 394, 416, 438, 460, 483, 505, 528, 551, 574, 597, 621, 644, 668, 691, 715, 739, 763, 787, 812, 836, 860, 885, 910, 934

Offset: 1

Mohammad K. Azarian, May 31 2012

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

Cf. A050502, A050504, A212455, A212456, A212458, A212459, A212460, A212461, A212462.

PROG(y := [], x := 100, LOOP(IF(x = 0, RETURN y), y := ADJOIN(FLOOR(4·x·LOG(4·x)), y), x := x - 1))

[Floor(4*n*Log(4*n)): n in [1..80]]; // Vincenzo Librandi, Feb 13 2013

Table[Floor[4*n*Log[4*n]], {n, 80}] (* Vincenzo Librandi, Feb 13 2013 *)

a(n)=4*n*log(4*n)\1 \\ Charles R Greathouse IV, Sep 04 2015

A212458 a(n) = floor(5n*log(5n)).

Original entry on oeis.org

8, 23, 40, 59, 80, 102, 124, 147, 171, 195, 220, 245, 271, 297, 323, 350, 377, 404, 432, 460, 488, 517, 545, 574, 603, 632, 662, 691, 721, 751, 781, 812, 842, 873, 903, 934, 965, 996, 1028, 1059, 1091, 1122, 1154, 1186

Offset: 1

Mohammad K. Azarian, May 31 2012

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

Cf. A050502, A050504, A212455, A212456, A212457, A212459, A212460, A212461, A212462.

PROG(y := [], x := 100, LOOP(IF(x = 0, RETURN y), y := ADJOIN(FLOOR(5·x·LOG(5·x)), y), x := x - 1))

[Floor(5*n*Log(5*n)): n in [1..80]]; // Vincenzo Librandi, Feb 13 2013

Table[Floor[5*n*Log[5*n]], {n, 80}] (* Vincenzo Librandi, Feb 13 2013 *)

a(n)=5*n*log(5*n)\1 \\ Charles R Greathouse IV, Sep 04 2015

A212459 a(n) = ceiling(2n*log(2n)).

Original entry on oeis.org

2, 6, 11, 17, 24, 30, 37, 45, 53, 60, 69, 77, 85, 94, 103, 111, 120, 130, 139, 148, 157, 167, 177, 186, 196, 206, 216, 226, 236, 246, 256, 267, 277, 287, 298, 308, 319, 330, 340, 351, 362, 373, 384, 395, 405, 417, 428

Offset: 1

Mohammad K. Azarian, May 31 2012

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

Cf. A050502, A050504, A212455, A212456, A212457, A212458, A212460, A212461, A212462.

PROG(y := [], x := 100, LOOP(IF(x = 0, RETURN y), y := ADJOIN(CEILING(2·x·LOG(2·x)), y), x := x - 1))

[Ceiling(2*n*Log(2*n)): n in [1..80]]; // Vincenzo Librandi, Feb 13 2013

Table[Ceiling[2*n*Log[2*n]], {n, 80}] (* Vincenzo Librandi, Feb 13 2013 *)

a(n) = ceil(2*n*log(2*n)); \\ Michel Marcus, Jan 11 2016

a(n) = A050502(2*n). - Michel Marcus, Jan 11 2016

A212460 a(n) = ceiling(3n*log(3n)).

Original entry on oeis.org

4, 11, 20, 30, 41, 53, 64, 77, 89, 103, 116, 130, 143, 157, 172, 186, 201, 216, 231, 246, 262, 277, 293, 308, 324, 340, 356, 373, 389, 405, 422, 439, 455, 472, 489, 506, 523, 540, 558, 575, 592, 610, 627, 645, 663, 680

Offset: 1

Mohammad K. Azarian, May 31 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Amya Luo, Pattern Avoidance in Nonnesting Permutations, Undergraduate Thesis, Dartmouth College (2024). See p. 16.

Cf. A050502, A050504, A212455, A212456, A212457, A212458, A212459, A212461, A212462.

PROG(y := [], x := 100, LOOP(IF(x = 0, RETURN y), y := ADJOIN(CEILING(3·x·LOG(3·x)), y), x := x - 1))

[Ceiling(3*n*Log(3*n)): n in [1..80]]; // Vincenzo Librandi, Feb 13 2013

Table[Ceiling[3*n*Log[3*n]], {n, 80}] (* Vincenzo Liobrandi, Feb 13 2013 *)

a(n) = ceil(3*n*log(3*n)); \\ Michel Marcus, Jan 11 2016

a(n) = A050502(3*n). - Michel Marcus, Jan 11 2016

A212461 a(n) = ceiling(4n*log(4n)).

Original entry on oeis.org

6, 17, 30, 45, 60, 77, 94, 111, 130, 148, 167, 186, 206, 226, 246, 267, 287, 308, 330, 351, 373, 395, 417, 439, 461, 484, 506, 529, 552, 575, 598, 622, 645, 669, 692, 716, 740, 764, 788, 813, 837, 861, 886, 911, 935

Offset: 1

Mohammad K. Azarian, May 31 2012

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

Cf. A050502, A050504, A212455, A212456, A212457, A212458, A212459, A212460, A212462.

PROG(y := [], x := 100, LOOP(IF(x = 0, RETURN y), y := ADJOIN(CEILING(4·x·LOG(4·x)), y), x := x - 1))

[Ceiling(4*n*Log(4*n)): n in [1..80]]; // Vincenzo Librandi, Feb 13 2013

Table[Ceiling[4*n*Log[4*n]], {n, 80}] (* Vincenzo Librandi, Feb 13 2013 *)

a(n) = ceil(4*n*log(4*n)); \\ Michel Marcus, Jan 11 2016

a(n) = A050502(4*n). - Michel Marcus, Jan 11 2016

A212462 a(n) = ceiling(5n*log(5n)).

Original entry on oeis.org

9, 24, 41, 60, 81, 103, 125, 148, 172, 196, 221, 246, 272, 298, 324, 351, 378, 405, 433, 461, 489, 518, 546, 575, 604, 633, 663, 692, 722, 752, 782, 813, 843, 874, 904, 935, 966, 997, 1029, 1060, 1092, 1123, 1155

Offset: 1

Mohammad K. Azarian, May 31 2012

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

Cf. A050502, A050504, A212455, A212456, A212457, A212458, A212459, A212460, A212461.

PROG(y := [], x := 100, LOOP(IF(x = 0, RETURN y), y := ADJOIN(CEILING(5·x·LOG(5·x)), y), x := x - 1))

[Ceiling(5*n*Log(5*n)): n in [1..80]]; // Vincenzo Librandi, Feb 14 2013

Table[Ceiling[5*n*Log[5*n]], {n, 80}] (* Vincenzo Librandi, Feb 14 2013 *)

a(n) = ceil(5*n*log(5*n)); \\ Michel Marcus, Jan 11 2016

a(n) = A050502(5*n). - Michel Marcus, Jan 11 2016

A050503 Nearest integer to n*log(n).

Original entry on oeis.org

0, 1, 3, 6, 8, 11, 14, 17, 20, 23, 26, 30, 33, 37, 41, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 85, 89, 93, 98, 102, 106, 111, 115, 120, 124, 129, 134, 138, 143, 148, 152, 157, 162, 167, 171, 176, 181, 186, 191, 196, 201, 205, 210, 215, 220, 225, 230, 236, 241

Offset: 1

N. J. A. Sloane, Dec 27 1999

nonn

The prime number theorem states that the n-th prime is asymptotic to n*log(n).

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Theorem 8.

Cf. A000040, A050502, A050504.

Table[Round[n Log[n]],{n,100}] (* Harvey P. Dale, Sep 27 2023 *)

A135736 Nearest integer to n*Sum_{k=1..n} 1/k = rounded expected coupon collection numbers.

Original entry on oeis.org

0, 1, 3, 6, 8, 11, 15, 18, 22, 25, 29, 33, 37, 41, 46, 50, 54, 58, 63, 67, 72, 77, 81, 86, 91, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 161, 166, 171, 176, 182, 187, 192, 198, 203, 209, 214, 219, 225, 230, 236, 242, 247, 253, 258, 264, 269, 275

Offset: 0

M. F. Hasler, Nov 29 2007

Somewhat more realistic than A052488 but more optimistic than A060293, the expected number of boxes that must be bought to get the full collection of N objects, if each box contains any one of them at random. See also comments in A060293, A052488.

			a(0) = 0 since nothing needs to be bought if nothing is to be collected.
a(1) = 1 since only 1 box needs to be bought if only 1 object is to be collected.
a(2) = 3 since the chance of getting the other object at the second purchase is only 1/2, so it takes 2 boxes on the average to get it.
a(3) = 6 since the chance of getting a new object at the second purchase is 2/3 so it takes 3/2 boxes in the mean, then the chance becomes 1/3 to get the 3rd, i.e., 3 other boxes on the average to get the full collection and the rounded value of 1 + 3/2 + 3 = 11/2 = 5.5 is 6.

Adam Hugill, Table of n, a(n) for n = 0..10000 (terms 0..999 from G. C. Greubel)

Cf. A060293, A052488, A050502-A050504.

seq(round(n*harmonic(n)), n=1..100); # Robert Israel, Oct 31 2016

Table[Round[n*Sum[1/k, {k, 1, n}]], {n,0,25}] (* G. C. Greubel, Oct 29 2016 *)

PARI
```
A135736(n)=round(n*sum(i=1,n,1/i))
```

n=100 #set the number of terms you want to calculate here
ans=0
finalans = []
finalans.append(0) #continuity with A135736
for i in range(1, n+1):
    ans+=(1/i)
    finalans.append(int(round(ans*i)))
print(finalans)
# Adam Hugill, Feb 14 2022

a(n) = round(n*A001008(n)/A002805(n)) = either A052488(n) or A060293(n).
a(n) ~ A060293(n) ~ A052488(n) ~ A050502(n) ~ A050503(n) ~ A050504(n) (asymptotically)
Conjecture: a(n) = round(n*(log(n) + gamma) + 1/2) for n > 0, where gamma = A001620. - Ilya Gutkovskiy, Oct 31 2016
The conjecture is false: a(2416101) = 36905656 while round(2416101*(log(2416101) + gamma) + 1/2) = 36905657, with the unrounded numbers being 36905656.499999982... and 36905656.500000016.... Heuristically this should happen infinitely often. - Charles R Greathouse IV, Oct 31 2016

cp's OEIS Frontend

A212455 a(n) = floor(2n*log(2n)).

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A212456 a(n) = floor(3n*log(3n)).

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A212457 a(n) = floor(4n*log(4n)).

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A212458 a(n) = floor(5n*log(5n)).

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A212459 a(n) = ceiling(2n*log(2n)).

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A212460 a(n) = ceiling(3n*log(3n)).

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A212461 a(n) = ceiling(4n*log(4n)).

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