A360940 Numbers k such that k / A000005(k) + k / A000010(k) is an integer.

1, 2, 3, 8, 10, 12, 18, 21, 24, 28, 36, 72, 78, 96, 108, 126, 128, 168, 224, 243, 288, 294, 384, 392, 756, 864, 930, 972, 1000, 1152, 1323, 1350, 1944, 2310, 2430, 2530, 2808, 3087, 3456, 4116, 6144, 6912, 7776, 10206, 10584, 13122, 13230, 13500, 13608, 18432

Offset: 1

Author

Ctibor O. Zizka, Feb 26 2023

Keywords

• nonn

Comments

It seems that odd k's {1, 3, 21, 243, 1323, 3087, ...} are relatively rare. A235353 is a subsequence of this sequence.

This sequence is infinite since A058891 is a subsequence. - Amiram Eldar, Feb 26 2023

Examples

			k = 1: 1 / A000005(1) + 1 / A000010(1) = 2, thus k = 1 is a term.
k = 2: 2 / A000005(2) + 2 / A000010(2) = 3, thus k = 2 is a term.
k = 3: 3 / A000005(3) + 3 / A000010(3) = 3, thus k = 3 is a term.
and so on.
		

Crossrefs

Cf. A000005, A000010, A058891, A235353.

Programs

Mathematica

Select[Range[10^4], IntegerQ[#/DivisorSigma[0, #] + #/EulerPhi[#]] &] (* Amiram Eldar, Feb 26 2023 *)

Python

from math import prod
from itertools import count, islice
from sympy import factorint
def A360940_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(startvalue,1)):
        f = factorint(k)
        t = prod(p**(e-1)*(p-1) for p, e in f.items())
        s = prod(e+1 for e in f.values())
        if not k*(s+t)%(s*t):
            yield k
A360940_list = list(islice(A360940_gen(),20)) # Chai Wah Wu, Mar 14 2023