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A342665 Numbers k for which phi(k)+1 is a multiple of d(k), where phi is Euler totient function (A000010) and d(n) gives the number of divisors of n (A000005).

%I A342665 #25 Apr 01 2021 06:07:28
%S A342665 1,2,4,25,81,121,289,529,841,1681,2209,2809,3481,5041,6889,7921,10201,
%T A342665 11449,12100,12769,17161,18769,22201,27889,28561,28900,29929,32041,
%U A342665 36481,38809,51529,54289,57121,63001,66049,69169,72361,78961,84100,85849,96721,100489,120409,124609,128881,146689,151321,160801,175561
%N A342665 Numbers k for which phi(k)+1 is a multiple of d(k), where phi is Euler totient function (A000010) and d(n) gives the number of divisors of n (A000005).
%C A342665 Numbers k such that A124331(k) = k. This is also a subsequence of the records of A124331 (both their values and their positions).
%C A342665 Terms other than 2 are a perfect square. Proof: phi(k) is even for k > 2. So phi(k)+1 is odd for k > 2. d(k) is odd only if k is a perfect square. So for any term k > 2 we need k to be a perfect square. Checking cases <= 2 leaves only 2 as the nonsquare in this sequence. - _David A. Corneth_, Mar 31 2021
%H A342665 David A. Corneth, <a href="/A342665/b342665.txt">Table of n, a(n) for n = 1..10000</a>
%t A342665 Select[Join[{1, 2}, Range[2, 420]^2], Divisible[EulerPhi[#] + 1, DivisorSigma[0, #]] &] (* _Amiram Eldar_, Mar 31 2021 *)
%o A342665 (PARI) isA342665(n) = !((eulerphi(n)+1) % numdiv(n));
%Y A342665 Cf. A000005, A000010, A020491.
%Y A342665 Fixed points of A124331. After 1, a subsequence of A015733.
%K A342665 nonn
%O A342665 1,2
%A A342665 _Antti Karttunen_, Mar 30 2021