A050499 -id:A050499 - cp's OEIS frontend

A330823 a(1) = 1; for n > 1, a(n) = a(n-1) - n if n is prime, otherwise a(n) = a(n-1) + floor(n/(log(n)-1)).

1, -1, -4, 6, 1, 8, 1, 8, 15, 22, 11, 19, 6, 14, 22, 31, 14, 23, 4, 14, 24, 34, 11, 22, 33, 44, 55, 67, 38, 50, 19, 31, 44, 57, 70, 83, 46, 60, 74, 88, 47, 62, 19, 34, 50, 66, 19, 35, 51, 68, 85, 102, 49, 67, 85, 103, 121, 139, 80, 99, 38, 57, 77, 97, 117, 137, 70

Offset: 1

Scott R. Shannon, Jan 02 2020

The Prime Number Theorem shows that the probability of a random number not greater than x being prime is approximately 1/log(x), therefore the probability of a number being composite in the same range is approximately (log(x)-1)/log(x). As this sequence subtracts n from the previous term if n is prime, or adds n with a weighting of 1/(log(n)-1) if n is composite, its expected value as n goes to infinity is approximately n*(1/(log(n)-1))*((log(n)-1)/log(n)) - n*(1/log(n)) = 0. We therefore expect that a(n)/n approaches 0 as n goes to infinity.

In the first 2 million terms the sequence changes sign 1900 times, has a maximum positive value of 160213275 at a(1772200), and a maximum negative value of -29535301 at a(1513751). The majority of terms are positive. See the image link below.

Scott R. Shannon, Graph showing a(n) for n = 1 to 2000000. The blue line is the x axis.
Wikipedia, Prime Number Theorem and Prime Counting Function.

Cf. A000040, A057835, A060769, A060770, A050499, A085581.

a[1] = 1; a[n_] := a[n] = a[n - 1] + If[PrimeQ[n], -n, Floor[n/(Log[n] - 1)]]; Array[a, 67] (* Amiram Eldar, Jan 05 2020 *)

A380198 Difference between pi(2^n) and the integer nearest to 2^n / log(2^n).

Original entry on oeis.org

-2, -1, 0, 0, 2, 3, 5, 8, 15, 24, 40, 72, 119, 212, 360, 633, 1128, 1989, 3580, 6386, 11537, 20897, 37980, 69354, 127336, 234054, 431877, 799754, 1484440, 2763961, 5156791, 9644970, 18080775, 33959344, 63902732, 120474951, 227515953, 430345298, 815241632

Offset: 1

James C. McMahon, Jan 16 2025

sign

			n   2^n   pi(2^n)  round(2^n/log(2^n))  a(n)
------------------------------------------------
1     2     1         3                  -2
2     4     2         3                  -1
3     8     4         4                   0
4    16     6         6                   0

Michael De Vlieger, Table of n, a(n) for n = 1..92

Cf. A000720, A007053, A050499, A053622, A057835.

Table[PrimePi[2^n]-Round[2^n/Log[2^n]],{n,39}]

a(n) = - A053622(2^n).

a(n) = A007053(n) - A050499(2^n).

A342355 Decimal expansion of 163/log(163).

Original entry on oeis.org

3, 1, 9, 9, 9, 9, 9, 8, 7, 3, 8, 4, 9, 0, 0, 8, 2, 6, 7, 5, 7, 5, 8, 3, 9, 3, 0, 2, 6, 5, 5, 6, 5, 4, 7, 9, 4, 1, 0, 9, 0, 6, 5, 1, 4, 9, 2, 0, 8, 2, 9, 3, 9, 6, 9, 6, 4, 0, 9, 9, 0, 9, 6, 6, 9, 6, 3, 1, 9, 5, 7, 6, 8, 4, 6, 6, 0, 8, 3, 2, 2, 1, 1, 7, 1, 2, 9, 5, 9, 5, 8, 9, 1, 8, 4, 9, 0

Offset: 2

Michal Paulovic, Mar 08 2021

A near-integer close to 32.

			31.9999987384900826...

MathOverflow, Why Is 163/ln(163) a Near-Integer?
MathOverflow, When is n/ln(n) close to an integer?
Index entries for transcendental numbers

Cf. A050499, A131920, A178805.

MATLAB
```
format long; 163 / log(163)
```
Maple
```
Digits:=100; evalf(163/ln(163));
```
Mathematica
```
RealDigits[163/Log[163], 10, 100][[1]]
```

default(realprecision, 100); 163 / log(163)

Equals 163/log(163).

cp's OEIS Frontend

A330823 a(1) = 1; for n > 1, a(n) = a(n-1) - n if n is prime, otherwise a(n) = a(n-1) + floor(n/(log(n)-1)).

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A380198 Difference between pi(2^n) and the integer nearest to 2^n / log(2^n).

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A342355 Decimal expansion of 163/log(163).

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