A038605 -id:A038605 - cp's OEIS frontend

A079415 a(n) = floor(prime(n)/n) * ceiling(prime(n)/n) / 2.

2, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 15, 10, 10, 10, 15, 15, 15, 15, 15, 15, 15

Offset: 1

Reinhard Zumkeller, Jan 07 2003

nonn

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

Cf. A038605, A079416, A000040.

[Floor(NthPrime(n)/n)*Ceiling(NthPrime(n)/n)/2: n in [1..80]]; // G. C. Greubel, Jan 19 2019

fc[n_]:=Module[{c=Prime[n]/n},(Floor[c]Ceiling[c])/2]; Array[fc,80] (* Harvey P. Dale, May 20 2015 *)

vector(80, n, floor(prime(n)/n)*ceil(prime(n)/n)/2) \\ G. C. Greubel, Jan 19 2019

[floor(nth_prime(n)/n)*ceil(nth_prime(n)/n)/2 for n in (1..80)] # G. C. Greubel, Jan 19 2019

A134919 Floor(n^(5/3)).

Original entry on oeis.org

1, 3, 6, 10, 14, 19, 25, 32, 38, 46, 54, 62, 71, 81, 91, 101, 112, 123, 135, 147, 159, 172, 186, 199, 213, 228, 243, 258, 273, 289, 305, 322, 339, 356, 374, 392, 410, 429, 448, 467, 487, 507, 527, 548, 569, 590, 612, 633, 656

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, A134918.

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

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

A347342 a(n) = prime(n) mod floor(prime(n) / n).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 3, 3, 1, 3, 1, 3, 1, 3, 3, 1, 3, 1, 3, 1, 1, 3, 3, 3, 3, 1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 3, 1, 3, 3, 1, 1, 3, 1, 2, 1, 1, 3, 2, 3, 4, 3, 4, 2, 1, 4, 4, 1, 1, 3, 4, 3

Offset: 1

Simon Strandgaard, Aug 27 2021

nonn

			a(1) =  2 mod floor( 2 / 1) =  2 mod 2 = 0,
a(2) =  3 mod floor( 3 / 2) =  3 mod 1 = 0,
a(3) =  5 mod floor( 5 / 3) =  5 mod 1 = 0,
a(4) =  7 mod floor( 7 / 4) =  7 mod 1 = 0,
a(5) = 11 mod floor(11 / 5) = 11 mod 2 = 1.

Simon Strandgaard, Plot of 1000 terms

Cf. A000040, A038605.

A347342[n_] := Mod[Prime[n] , Floor[Prime[n]/n]]; Table[A347342[n], {n, 1, 86}] (* Robert P. P. McKone, Aug 27 2021 *)

PARI
```
a(n) = prime(n) % (prime(n) \ n);
```

require 'prime'
values = []
Prime.first(30).each_with_index do |prime,i|
    values << prime % (prime/(i+1))
end
p values

a(n) = A000040(n) mod A038605(n).

A066367 The floor(prime(n)/n)-perfect numbers, where prime(n) denotes the n-th prime and f-perfect numbers for an arithmetical function f are defined in A066218.

Original entry on oeis.org

5, 7, 11, 49, 169

Offset: 1

Joseph L. Pe, Dec 21 2001

There do not seem to be any more terms. There are no terms between 170 and 10^5.

J. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.

Cf. A038605, A066218.

f[x_] := Floor[Prime[x] / x]; Select[ Range[2, 10^3], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]

A308082 Numbers k such that floor(prime(k)/k) < floor(prime(k+1)/(k+1)).

Original entry on oeis.org

4, 11, 30, 68, 72, 180, 189, 442, 1051, 1059, 2700, 6454, 6458, 6465, 6472, 15927, 40072, 40121, 100361, 100363, 251706, 251709, 251723, 251737, 251761, 637234, 637320, 637323, 637330, 637340, 1617174, 4124436, 4124466, 4124472, 4124705, 10553414

Offset: 1

Giorgos Kalogeropoulos, May 11 2019

nonn

Largest k below 10^8 is 69709965.

If instead of "less than" in floor(prime(k)/k) < floor(prime(k+1)/(k+1)), we use "greater than", we get A283053.

Cf. A000040, A038605, A283053.

Select[Range@100000, Floor[Prime@#/#] < Floor[Prime[# + 1]/(# + 1)] &]

isok(k) = prime(k)\k < prime(k+1)\(k+1); \\ Michel Marcus, May 11 2019

lista(nn) = {my(p=2, ip=1, q=3); for (n=1, nn, if (p\ip < q\(ip+1), print1(ip, ", ")); p = q; ip ++; q = nextprime(p+1););} \\ Michel Marcus, May 11 2019

cp's OEIS Frontend

A079415 a(n) = floor(prime(n)/n) * ceiling(prime(n)/n) / 2.

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Sage

A134919 Floor(n^(5/3)).

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Magma

Mathematica

A347342 a(n) = prime(n) mod floor(prime(n) / n).

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Formula

A066367 The floor(prime(n)/n)-perfect numbers, where prime(n) denotes the n-th prime and f-perfect numbers for an arithmetical function f are defined in A066218.

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Mathematica

A308082 Numbers k such that floor(prime(k)/k) < floor(prime(k+1)/(k+1)).

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Comments

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Mathematica

PARI

PARI