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

%I A360940 #19 Mar 14 2023 11:46:03
%S A360940 1,2,3,8,10,12,18,21,24,28,36,72,78,96,108,126,128,168,224,243,288,
%T A360940 294,384,392,756,864,930,972,1000,1152,1323,1350,1944,2310,2430,2530,
%U A360940 2808,3087,3456,4116,6144,6912,7776,10206,10584,13122,13230,13500,13608,18432
%N A360940 Numbers k such that k / A000005(k) + k / A000010(k) is an integer.
%C A360940 It seems that odd k's {1, 3, 21, 243, 1323, 3087, ...} are relatively rare. A235353 is a subsequence of this sequence.
%C A360940 This sequence is infinite since A058891 is a subsequence. - _Amiram Eldar_, Feb 26 2023
%e A360940 k = 1: 1 / A000005(1) + 1 / A000010(1) = 2, thus k = 1 is a term.
%e A360940 k = 2: 2 / A000005(2) + 2 / A000010(2) = 3, thus k = 2 is a term.
%e A360940 k = 3: 3 / A000005(3) + 3 / A000010(3) = 3, thus k = 3 is a term.
%e A360940 and so on.
%t A360940 Select[Range[10^4], IntegerQ[#/DivisorSigma[0, #] + #/EulerPhi[#]] &] (* _Amiram Eldar_, Feb 26 2023 *)
%o A360940 (Python)
%o A360940 from math import prod
%o A360940 from itertools import count, islice
%o A360940 from sympy import factorint
%o A360940 def A360940_gen(startvalue=1): # generator of terms >= startvalue
%o A360940     for k in count(max(startvalue,1)):
%o A360940         f = factorint(k)
%o A360940         t = prod(p**(e-1)*(p-1) for p, e in f.items())
%o A360940         s = prod(e+1 for e in f.values())
%o A360940         if not k*(s+t)%(s*t):
%o A360940             yield k
%o A360940 A360940_list = list(islice(A360940_gen(),20)) # _Chai Wah Wu_, Mar 14 2023
%Y A360940 Cf. A000005, A000010, A058891, A235353.
%K A360940 nonn
%O A360940 1,2
%A A360940 _Ctibor O. Zizka_, Feb 26 2023