A005394 -id:A005394 - cp's OEIS frontend

A056772 Numbers k such that phi(k+4) = phi(k) + 4, where phi(k) = A000010(k) is Euler's totient function.

3, 7, 12, 13, 18, 19, 24, 28, 36, 37, 40, 43, 66, 67, 79, 88, 97, 103, 109, 124, 127, 163, 184, 193, 223, 229, 232, 277, 307, 313, 328, 349, 379, 397, 424, 439, 457, 463, 487, 499, 508, 613, 643, 664, 673, 712, 739, 757, 769, 823, 853, 859, 877, 883, 904, 907

Offset: 1

Labos Elemer, Aug 17 2000

nonn

In contrast with A015913, composite solutions are not rare. Prime solutions are common.
From Kevin J. Gomez, Mar 02 2016: (Start)
Composite solutions have two known forms:
  n such that n = 4 * (2^p - 1) where 2^p - 1 is a Mersenne prime. (A001348)
  n such that n = 8q where q is a Sophie Germain prime. (A005394)
There are composite solutions (such as 36) that do not fit either of these forms. (End)

			For k = 1048: phi(1048) = 520, phi(1048+4) = 524.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Seiichi Manyama)

Cf. A000010, A015913 (sigma(k+4) = sigma(k) + 4).

Cf. A001838 (k=2), this sequence (k=4), A262084 (k=6), A262085 (k=8), A262086 (k=10).

[n: n in [1..1000] | EulerPhi(n+4) eq EulerPhi(n)+4]; // Vincenzo Librandi, Sep 11 2015

Select[Range@1000, EulerPhi@(# + 4)== EulerPhi[#] + 4 &] (* Vincenzo Librandi, Sep 11 2015 *)
Position[Partition[EulerPhi[Range[1000]],5,1],?(#[[1]]+4==#[[5]]&),1, Heads-> False]//Flatten (* _Harvey P. Dale, Dec 18 2019 *)

isok(n) = eulerphi(n+4) == eulerphi(n) + 4; \\ Michel Marcus, Sep 11 2015

A090583 Gosper's approximation to n!, sqrt((2n+1/3)Pi)*n^n/e^n, rounded to nearest integer.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5039, 40316, 362851, 3628561, 39914615, 478979481, 6226774954, 87175314872, 1307635379670, 20922240412500, 355679137660826, 6402240370021199, 121642823201649058, 2432860847996122437, 51090157192742729183, 1123984974735953018069

Offset: 0

Hugo Pfoertner, Jan 10 2004

nonn

G. C. Greubel, Table of n, a(n) for n = 0..449 (terms 0..200 from Alois P. Heinz)
Peter Luschny, Approximations to the Factorial Function.
Eric Weisstein's World of Mathematics, Stirling's Approximation.

Cf. A000142, A005394, A055775.

C := ComplexField(); [Round(Sqrt((2*n + 1/3)*Pi(C))*n^n/Exp(n)): n in [0..30]]; // G. C. Greubel, Nov 28 2017

Maple

Digits:= 2000; a:= n-> round(sqrt((2*n+1/3)*Pi)*n^n/exp(n)): seq(a(n), n=0..30); # Alois P. Heinz, Feb 04 2013

Mathematica

Join[{1}, Table[Round[Sqrt[(2*n + 1/3)*Pi]*n^n/Exp[n]], {n, 1, 50}]] (* G. C. Greubel, Nov 28 2017 *)

PARI

a(n) = round(sqrt((2*n+1/3)*Pi)*n^n/exp(n)); \\ Bill McEachen, Aug 16 2014

A260657 Rounded error in Stirling's formula: a(n) = round(n! - exp(-n)n^(n+1/2)sqrt(2*Pi)).

Original entry on oeis.org

1, 0, 0, 0, 0, 2, 10, 60, 418, 3343, 30104, 301175, 3314114, 39781325, 517289459, 7243645801, 108675472777, 1739099429899, 29569079533691, 532313816538037, 10115161415506606, 202324846199795597, 4249233149373416698, 93491368355657653179, 2150474710445177712523

Offset: 0

Vladimir Reshetnikov, Nov 13 2015

nonn

G. C. Greubel, Table of n, a(n) for n = 0..200

Cf. A000142, A005394.

a:= n-> n!-round(sqrt(2*Pi*n)*(n/exp(1))^n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 24 2024

Table[Round[n! - Exp[-n] n^(n+1/2) Sqrt[2 Pi]], {n, 0, 24}]

def a(n): # Throws an error if result could not be computed exactly.
    rif = RealIntervalField(max(4,10*n))
    r = rif(factorial(n)-(n^(1/2+n)*sqrt(2*pi))/exp(n))
    return r.unique_round()
for n in (0..100): print(n, a(n)) # b-file style; Peter Luschny, Nov 18 2015

a(n) ~ exp(-n)*n^(n-1/2)*sqrt(2*Pi)/12.

a(n) = A000142(n) - A005394(n). - Alois P. Heinz, Jan 24 2024

A127585 Exponential error term from Stirling's Approximation.

Original entry on oeis.org

1, 1, 18, 345, 10243, 437769, 25260317, 1873346813, 172254143084, 19114537903943, 2506628271002200, 382005168783773474, 66734799966312471195, 13212509243902296154744, 2936153006332857671962341, 726345521215072990990045577, 198595552305314906351047196508

Offset: 0

Jonathan Vos Post, Apr 02 2007

			a(1) = Floor[(sqrt(2*pi) * (1^1) * (1^(1/2))) - 1! ] = Floor(1.50662827) = 1.
a(2) = Floor[(sqrt(2*pi) * (2^2) * (2^(2/2))) - 2! ] = Floor(18.0530262) = 18.

Eric Weisstein's World of Mathematics, Stirling's Series.
Eric Weisstein's World of Mathematics, Stirling's Approximation.

Cf. A005394, A046968, A046969, A055775, A127426.

a(n) = floor(sqrt(2*Pi)*(n^n)*(n^(n/2))) - n!.

More terms from Alois P. Heinz, Jan 24 2024

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A056772 Numbers k such that phi(k+4) = phi(k) + 4, where phi(k) = A000010(k) is Euler's totient function.

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A090583 Gosper's approximation to n!, sqrt((2*n+1/3)*Pi)*n^n/e^n, rounded to nearest integer.

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A260657 Rounded error in Stirling's formula: a(n) = round(n! - exp(-n)*n^(n+1/2)*sqrt(2*Pi)).

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A127585 Exponential error term from Stirling's Approximation.

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A090583 Gosper's approximation to n!, sqrt((2n+1/3)Pi)*n^n/e^n, rounded to nearest integer.

A260657 Rounded error in Stirling's formula: a(n) = round(n! - exp(-n)n^(n+1/2)sqrt(2*Pi)).