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A359059
Numbers k such that phi(k) + rad(k) + psi(k) is a multiple of 3.
1
1, 2, 3, 5, 7, 8, 9, 11, 13, 17, 18, 19, 20, 23, 27, 29, 31, 32, 36, 37, 41, 42, 43, 44, 45, 47, 49, 50, 53, 54, 59, 61, 63, 67, 68, 71, 72, 73, 78, 79, 80, 81, 83, 84, 89, 90, 92, 97, 99, 101, 103, 105, 107, 108, 109, 110, 113, 114, 116, 117, 125, 126, 127, 128, 131, 135, 137, 139
OFFSET
1,2
COMMENTS
When k is prime (denote as p), phi(p) = p - 1, rad(p) = p, and psi(p) = p + 1, so phi(p) + rad(p) + psi(p) = 3*p. Therefore, A000040 is a subsequence.
When k = p^m (m>=1) with p prime, phi(p^m) = (p-1)*p^(m-1), rad(p^m) = p, and psi(p^m) = (p+1)*p^(m-1), so phi(p^m) + rad(p^m) + psi(p^m) = 2*p^m + p = p * (1+2*p^(m-1)). Then, this expression is a multiple of 3 iff p == 0 or 1 (mod 3), equivalently iff p is a generalized cuban prime of A007645. Therefore, as 1 is also a term, every sequence {p^m, p in A007645, m>=0} is a subsequence. See crossrefs section. - Bernard Schott, Jan 25 2023 after an observation of Alois P. Heinz
EXAMPLE
8 is a term because 4+2+12 is divisible by 3.
MATHEMATICA
q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Divisible[Times @@ ((p - 1)*p^(e - 1)) + Times @@ p + Times @@ ((p + 1)*p^(e - 1)), 3]]; Select[Range[170], q] (* Amiram Eldar, Dec 15 2022 *)
PROG
(Python)
from sympy.ntheory.factor_ import totient
from sympy import primefactors, prod
def rad(n): return 1 if n < 2 else prod(primefactors(n))
def psi(n):
plist = primefactors(n)
return n*prod(p+1 for p in plist)//prod(plist)
# Output display terms.
for n in range(1, 170):
if(0 == (totient(n) + rad(n) + psi(n)) % 3):
print(n, end = ", ")
(PARI) isok(m) = ((eulerphi(m) + factorback(factorint(m)[, 1]) + m*sumdiv(m, d, moebius(d)^2/d)) % 3) == 0; \\ Michel Marcus, Dec 27 2022
CROSSREFS
Cf. A000010 (phi), A000040, A001615 (psi), A007645, A007947 (rad), A001748 (3*p), A000244.
Subsequences of the form {p^n, n>=0}: A000244 (p=3), A000420 (p=7), A001022 (p=13), A001029 (p=19), A009975 (p=31), A009981 (p=37), A009987 (p=43).
Sequence in context: A063743 A353968 A144100 * A328320 A086539 A171217
KEYWORD
nonn
AUTHOR
Torlach Rush, Dec 14 2022
STATUS
approved