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A349225 Numbers k such that k | A002619(k). 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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, ...

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.

EXAMPLE

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

MATHEMATICA

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

PROG

(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

CROSSREFS

Cf. A000010 (phi), A002619.

Sequence in context: A350526 A173975 A349770 * A199884 A028331 A279729

Adjacent sequences: A349222 A349223 A349224 * A349226 A349227 A349228

KEYWORD

nonn

AUTHOR

Amiram Eldar, Nov 11 2021

STATUS

approved

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Last modified January 29 23:01 EST 2023. Contains 359939 sequences. (Running on oeis4.)