cp's OEIS Frontend

A359563 Odd numbers that have at least two divisors with the same value of the Euler totient function (A000010).

%I A359563 #12 Mar 05 2025 16:13:14
%S A359563 63,189,273,315,441,513,567,585,693,819,825,945,1071,1197,1323,1365,
%T A359563 1449,1539,1575,1701,1755,1827,1911,1953,2079,2107,2109,2205,2255,
%U A359563 2331,2457,2475,2565,2583,2709,2835,2925,2961,3003,3069,3075,3087,3213,3339,3465,3549
%N A359563 Odd numbers that have at least two divisors with the same value of the Euler totient function (A000010).
%C A359563 The even numbers are excluded from this sequence since every even number has this property: it is divisible by 1 and 2, and phi(1) = phi(2) = 1.
%C A359563 If k is a term then all the odd multiples of k are terms. The primitive terms are in A359564.
%C A359563 The numbers of terms below 10^k, for k = 1, 2, ..., are 0, 1, 12, 140, 1402, 14193, 142606, 1427749, 14283236, 142855925, ... . Apparently, the asymptotic density of this sequence exists and equals 0.01428... .
%C A359563 The least term that is not divisible by 3 is a(26) = 2107.
%H A359563 Amiram Eldar, <a href="/A359563/b359563.txt">Table of n, a(n) for n = 1..10000</a>
%e A359563 63 is a term since it is odd, 7 and 9 are both divisors of 63, and phi(7) = phi(9) = 6.
%t A359563 Select[Range[1, 3500, 2], !UnsameQ @@ EulerPhi[Divisors[#]] &]
%t A359563 Select[Range[1,3601,2],Max[Tally[EulerPhi[Divisors[#]]][[;;,2]]]>1&] (* _Harvey P. Dale_, Mar 05 2025 *)
%o A359563 (PARI) is(k) = k>1 && k%2 && numdiv(k) > #Set(apply(x->eulerphi(x), divisors(k)));
%Y A359563 Complement of A326835 within the odd numbers.
%Y A359563 Cf. A000010, A359564, A359565, A359566.
%K A359563 nonn
%O A359563 1,1
%A A359563 _Amiram Eldar_, Jan 06 2023