A050504 -id:A050504 - cp's OEIS frontend

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

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

A250621 a(n) = floor(n*log(prime(n))).

Original entry on oeis.org

0, 2, 4, 7, 11, 15, 19, 23, 28, 33, 37, 43, 48, 52, 57, 63, 69, 73, 79, 85, 90, 96, 101, 107, 114, 119, 125, 130, 136, 141, 150, 156, 162, 167, 175, 180, 187, 193, 199, 206, 212, 218, 225, 231, 237, 243, 251, 259, 265, 271, 278, 284, 290, 298, 305, 312, 318, 324, 331

Offset: 1

Freimut Marschner, Nov 26 2014

From n < prime(n), n >= 1 follows that n*log(n) < prime(n) < n*log(prime(n)), n >= 4. This inequality is included in the prime number theorem PNT.

			For n = 1, prime(1) = 2, floor(1*0.69... = 0.69...) = 0 ;
For n = 25, prime(25) = 97, floor(25*4.57... = 114.36...) = 114.

Cf. A050504 (floor(n*log(n))), A086861 (floor(prime(n)/log(prime(n)))), A085581 (floor(prime(n)/log(n))), A050504 (integer part of n*log(n)), A050503 (nearest integer to n*log(n)), A050502 (ceiling of n*log(n)).

Table[Floor[n Log[Prime[n]]],{n,60}] (* Harvey P. Dale, Aug 13 2019 *)

vector(100,n,floor(n*log(prime(n)))) \\ Derek Orr, Nov 28 2014

A088301 a(n) = p(n)/p(n-1), where p(n) = ( floor(nlog(n)) / Product_{j=2..pi(floor(nlog(n)))} prime(j) )!.

Original entry on oeis.org

1, 2, 4, 48, 90, 12, 3360, 18, 9240, 15600, 756, 31680, 42840, 59280, 1848, 99360, 6497400, 2970, 185136, 234360, 18670080, 347760, 421800, 480480, 557928, 55965360, 70073640, 857280, 98960400, 1157520, 11880, 162983520, 190578024

Offset: 2

Roger L. Bagula, Nov 04 2003

G. C. Greubel, Table of n, a(n) for n = 2..1000

Cf. A000720, A050504.

m:=50;
b:= [ #PrimesUpTo(n): n in [1..2+Floor(2*m*Log(2*m))] ];
f:= func< n | Factorial( Floor(n*Log(n)) )/(&*[ NthPrime(j): j in [2..b[Floor(n*Log(n))]] ]) >;
A088301:= func< n | n le 3 select n-1 else f(n)/f(n-1) >;
[A088301(n): n in [2..m]]; // G. C. Greubel, Dec 18 2022

p[n_]:=Factorial[Floor[n*Log[n]]]/ Product[Prime[i], {i, 2, PrimePi[Floor[n*Log[n]]]}];
Table[p[n]/p[n-1], {n,2,50}]

def p(n): return factorial( floor(n*log(n)) )/product(nth_prime(j) for j in (2..prime_pi(floor(n*log(n)))))
def A088301(n): return p(n)/p(n-1)
[A088301(n) for n in range(2,50)] # G. C. Greubel, Dec 18 2022

a(n) = p(n)/p(n-1), where p(n) = ( floor(n*log(n)) / Product_{j=2..pi(floor(n*log(n)))} prime(j) )!.

Edited by G. C. Greubel, Dec 18 2022

A156625 Floor(integral of log(x) from 1 to n).

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 9, 11, 14, 16, 18, 21, 23, 26, 29, 32, 35, 37, 40, 43, 47, 50, 53, 56, 59, 62, 66, 69, 73, 76, 79, 83, 86, 90, 94, 97, 101, 104, 108, 112, 115, 119, 123, 127, 131, 134, 138, 142

Offset: 2

Jack W Grahl, Feb 11 2009

Cf. A156624.

Table[Floor[Integrate[Log[n],{n,1,x}]],{x,2,50}] (* Harvey P. Dale, May 08 2023 *)

[floor(N(integral(ln(x), x, 1, n))) for n in range (2,50)]

a(n)=A050504(n)-n+1. - R. J. Mathar, Feb 12 2009

A212293 Integer part of n*log(n)^2.

Original entry on oeis.org

0, 0, 3, 7, 12, 19, 26, 34, 43, 53, 63, 74, 85, 97, 110, 122, 136, 150, 164, 179, 194, 210, 226, 242, 259, 275, 293, 310, 328, 347, 365, 384, 403, 422, 442, 462, 482, 502, 523, 544, 565, 586, 608, 630, 652, 674, 696, 719, 742, 765, 788, 811, 835, 859, 883, 907, 931, 956

Offset: 1

Charles R Greathouse IV, Jun 21 2012

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A050504.

Table[Floor[n Log[n]^2],{n,60}] (* Harvey P. Dale, Oct 24 2018 *)

PARI
```
a(n)=n*log(n)^2\1
```

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

A235707 a(n) = floor(n^2 * log(n)).

Original entry on oeis.org

0, 2, 9, 22, 40, 64, 95, 133, 177, 230, 290, 357, 433, 517, 609, 709, 818, 936, 1062, 1198, 1342, 1496, 1658, 1830, 2011, 2202, 2402, 2612, 2831, 3061, 3300, 3548, 3807, 4076, 4355, 4644, 4943, 5252, 5572, 5902, 6242, 6593, 6954, 7326, 7708, 8101, 8504, 8919

Offset: 1

Alex Ratushnyak, Apr 20 2014

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

Cf. A050504.

[Floor(n^2 * Log(n)): n in [1..50]]; // Vincenzo Librandi, Apr 22 2014

Table[Floor[n^2 Log[n]], {n, 1, 50}] (* Vincenzo Librandi, Apr 22 2014 *)

import math
for n in range(1, 100):  print(int(n*n*math.log(n)), end=', ')

A076829 a(n) = floor(log(log(n^n^n))).

Original entry on oeis.org

1, 3, 5, 8, 11, 14, 17, 20, 23, 27, 30, 34, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 90, 94, 98, 103, 107, 112, 116, 121, 125, 130, 134, 139, 144, 148, 153, 158, 163, 167, 172, 177, 182, 187, 192, 196, 201, 206, 211, 216, 221, 226, 231, 236, 241, 247

Offset: 2

Neil Fernandez, Nov 20 2002

nonn

			3^3^3 = 3^27 = 7625597484987. Log(log(7625597484987)) = 3.38..., so a(3)= 3.

Cf. A002488, A000312, A050504.

Table[ Floor[ n * Log[n] + Log[ Log[n]]], {n, 2, 60}]

a(n) = floor(n*log(n) + log(log(n))).

Edited by Robert G. Wilson v, Nov 22 2002

Undefined a(1) removed by Sean A. Irvine, Apr 17 2025

A217864 Number of prime numbers between floor(nlog(n)) and (n + 1)log(n + 1).

Original entry on oeis.org

0, 2, 2, 2, 0, 2, 1, 2, 2, 1, 1, 2, 0, 1, 2, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 0, 1, 1, 2, 1, 2, 1, 0, 0, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 1, 1, 0, 2, 2, 0, 0, 1, 0, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 2, 2, 0, 1, 0, 1, 3, 2, 0, 0, 1, 1, 0, 2, 1, 1, 0, 1, 1, 2, 1, 1

Offset: 1

Jon Perry, Oct 13 2012

nonn

Conjecture: a(n) is unbounded.
If Riemann Hypothesis is true, this is probably true as the PNT is generally a lower bound for Pi(n).
Conjecture: a(n)=0 infinitely often.
The first conjecture follows from Dickson's conjecture. The second conjecture follows from a theorem of Brauer & Zeitz on prime gaps. - Charles R Greathouse IV, Oct 15 2012

			log(1)=0 and 2*log(2) ~ 1.38629436112. Hence, a(1)=0.
Floor(2*log(2)) = 1 and 3*log(3) ~ 3.295836866. Hence, a(2)=2.

A. Brauer and H. Zeitz, Über eine zahlentheoretische Behauptung von Legendre, Sitz. Berliner Math. Gee. 29 (1930), pp. 116-125; cited in Erdos 1935.

Paul Erdős, On the difference of consecutive primes, Quart. J. Math., Oxford Ser. 6 (1935), pp. 124-128.

An alternate version of A166712.

Cf. A217564, A096509, A000905, A050504, A000720.

function isprime(i) {
if (i==1) return false;
if (i==2) return true;
if (i%2==0) return false;
for (j=3;j<=Math.floor(Math.sqrt(i));j+=2)
if (i%j==0) return false;
return true;
}
for (i=1;i<88;i++) {
c=0;
for (k=Math.floor(i*Math.log(i));k<=(i+1)*Math.log(i+1);k++) if (isprime(k)) c++;
document.write(c+", ");
}

Table[s = Floor[n*Log[n]]; PrimePi[(n+1) Log[n+1]] - PrimePi[s] + Boole[PrimeQ[s]], {n, 100}] (* T. D. Noe, Oct 15 2012 *)

a(n)=sum(k=n*log(n)\1,(n+1)*log(n+1),isprime(k)) \\ Charles R Greathouse IV, Oct 15 2012

A316681 a(n) = floor(n*log(n)/2).

Original entry on oeis.org

0, 0, 1, 2, 4, 5, 6, 8, 9, 11, 13, 14, 16, 18, 20, 22, 24, 26, 27, 29, 31, 34, 36, 38, 40, 42, 44, 46, 48, 51, 53, 55, 57, 59, 62, 64, 66, 69, 71, 73, 76, 78, 80, 83, 85, 88, 90, 92, 95, 97, 100, 102, 105, 107, 110, 112, 115, 117, 120, 122, 125, 127

Offset: 1

Marcos Librán López, Jul 10 2018

			a(1) = floor(1*log(1)/2) = 0;
a(2) = floor(2*log(2)/2) = 0.

Cf. A050504.

Array[Floor[#*Log[#]/2] &, 70] (* Robert G. Wilson v, Jul 21 2018 *)

a(n) = floor(n*log(n)/2); \\ Michel Marcus, Jul 13 2018

A378422 Primes of the form floor(k*log(k)).

Original entry on oeis.org

3, 5, 13, 19, 23, 29, 59, 97, 271, 281, 307, 313, 383, 421, 443, 449, 499, 557, 691, 709, 727, 733, 739, 751, 757, 769, 787, 947, 953, 971, 1009, 1021, 1091, 1097, 1103, 1129, 1231, 1237, 1283, 1289, 1321, 1367, 1373, 1399, 1439, 1511, 1531, 1571, 1597

Offset: 1

Tristan M. Phillips, Nov 25 2024

nonn

Primes corresponding to A066928. Primes in the sequence A050504.

cp's OEIS Frontend

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

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Maple

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A250621 a(n) = floor(n*log(prime(n))).

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Mathematica

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A088301 a(n) = p(n)/p(n-1), where p(n) = ( floor(n*log(n)) / Product_{j=2..pi(floor(n*log(n)))} prime(j) )!.

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Magma

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SageMath

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A156625 Floor(integral of log(x) from 1 to n).

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Mathematica

Sage

Formula

A212293 Integer part of n*log(n)^2.

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Mathematica

PARI

Formula

A235707 a(n) = floor(n^2 * log(n)).

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Magma

Mathematica

Python

A076829 a(n) = floor(log(log(n^n^n))).

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Examples

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Programs

Mathematica

Formula

Extensions

A217864 Number of prime numbers between floor(n*log(n)) and (n + 1)*log(n + 1).

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A088301 a(n) = p(n)/p(n-1), where p(n) = ( floor(nlog(n)) / Product_{j=2..pi(floor(nlog(n)))} prime(j) )!.

A217864 Number of prime numbers between floor(nlog(n)) and (n + 1)log(n + 1).