A349225 Numbers k such that k | A002619(k).

1, 6, 8, 19, 28, 30, 80, 93, 119, 126, 136, 156, 186, 192, 205, 312, 351, 384, 448, 483, 567, 774, 820, 896, 945, 1081, 1100, 1187, 1240, 1375, 1464, 2268, 2628, 2720, 2898, 3197, 3744, 3840, 4544, 4992, 5079, 6200, 6567, 7296, 7832, 9184, 12288, 12636, 16578

Offset: 1

Author

Amiram Eldar, Nov 11 2021

Keywords

• nonn

Comments

Chao (1982) proved that k | Sum_{d|k} phi(d)^2*d^(k/d-1)*(k/d-1)! for all k. The quotients are A002619(k). This sequence consists of numbers k such that this sum is divisible by k^2.

There are terms k such that k^2 | A002619(k): 1, 8, 1081, ...

Examples

			6 is a term since A002619(6) = 24 is divisible by 6.
		

References

• József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 3, p. 192.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..200
• Chong-Yun Chao, Generalizations of theorems of Wilson, Fermat and Euler, Journal of Number Theory, Vol. 15, No. 1 (1982), pp. 95-114.

Crossrefs

Cf. A000010 (phi), A002619.

Programs

Mathematica

f[n_] := DivisorSum[n, EulerPhi[#]^2 * #^(n/#) * (n/#)! &]/n^2; Select[Range[1000], Divisible[f[#], #] &]

Python

from itertools import count, islice
from sympy import divisors, totient, factorial
def A349225_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:not sum(totient(m:=n//d)**2*factorial(d)*m**d for d in divisors(n,generator=True)) % n**3, count(max(startvalue,1)))
A349225_list = list(islice(A349225_gen(),10)) # Chai Wah Wu, Nov 07 2022