cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Dennis Gruhlke

Dennis Gruhlke's wiki page.

Dennis Gruhlke has authored 2 sequences.

A350777 Numbers k where phi(k) divides k - 3.

Original entry on oeis.org

1, 2, 3, 9, 195, 5187, 1141967133868035, 3658018932844533311864835
Offset: 1

Author

Albert Böschow and Dennis Gruhlke, Jan 15 2022

Keywords

Comments

Numbers in this sequence larger than 2 have to be odd, since phi(n) is even for n > 2, so n - 3 cannot be odd. Therefore n itself must be odd.
Terms having (k-3)/phi(k) = 2 are shared with A226105. - Max Alekseyev, Oct 26 2023

Examples

			phi(195) = 96, 195 - 3 = 192, and 96 divides 192.
		

Crossrefs

Programs

  • Mathematica
    Select[Range[6000], Divisible[#-3, EulerPhi[#]] &] (* Amiram Eldar, Jan 19 2022 *)
  • PARI
    isok(k) = !((k-3) % eulerphi(k)); \\ Michel Marcus, Jan 19 2022
    
  • Python
    from sympy import totient
    print("1, 2", end=", ")
    for k in range (3, 10**8, 2):
        if (k-3)%totient(k)==0:
            print(k, end=", ", flush=True) # Martin Ehrenstein, Mar 26 2022

Extensions

a(7)-a(8) from Max Alekseyev, Nov 05 2023

A348183 a(n) is the determinant of the n X n matrix M = (m_{i,j}), i,j from 0 to n-1: m{i,j} = (i+j)^2 mod n.

Original entry on oeis.org

1, 0, -1, -2, 0, 250, 5616, -33614, 0, 204073344, -900000000, -9431790764, 0, 10752962364222, -1870899108384768, -36328974609375000, 0, 22899384412078526344, -111400529859275793629184, -43843094862278417487512, 0, 2870507605405055660542502550, 67015802375208384199755038720
Offset: 0

Author

Dennis Gruhlke and Albert Böschow, Oct 05 2021

Keywords

Comments

It seems that for values of n divisible by 4 -> a(n) = 0 and rank(M) = n/2.

Examples

			a(2) =  |0^2 mod 2, 1^2 mod 2| = -1
        |1^2 mod 2, 2^2 mod 2|
--
        |0^2 mod 3, 1^2 mod 3, 2^2 mod 3|
a(3) =  |1^2 mod 3, 2^2 mod 3, 3^2 mod 3| = -2
        |2^2 mod 3, 3^2 mod 3, 4^2 mod 3|
		

Crossrefs

Cf. A070896.

Programs

  • Mathematica
    a[n_]:=Table[Mod[(i+j)^2,n],{i,0,n-1},{j,0,n-1}]; Join[{1},Table[Det[a[n]], {n, 22}]] (* Stefano Spezia, Oct 06 2021 *)
  • PARI
    a(n) = matdet(matrix(n, n, i, j, i--; j--; (i+j)^2 % n)); \\ Michel Marcus, Oct 06 2021
    
  • Python
    from sympy import Matrix
    def A348183(n): return Matrix(n,n,[pow(i+j,2,n) for i in range(n) for j in range(n)]).det() # Chai Wah Wu, Nov 24 2021

Extensions

a(14)-a(17) corrected by and more terms from Stefano Spezia, Oct 06 2021.