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).

1, 2, 4, 25, 81, 121, 289, 529, 841, 1681, 2209, 2809, 3481, 5041, 6889, 7921, 10201, 11449, 12100, 12769, 17161, 18769, 22201, 27889, 28561, 28900, 29929, 32041, 36481, 38809, 51529, 54289, 57121, 63001, 66049, 69169, 72361, 78961, 84100, 85849, 96721, 100489, 120409, 124609, 128881, 146689, 151321, 160801, 175561

Offset: 1

Author

Antti Karttunen, Mar 30 2021

Keywords

• nonn

Comments

Numbers k such that A124331(k) = k. This is also a subsequence of the records of A124331 (both their values and their positions).

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

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000010, A020491.

Fixed points of A124331. After 1, a subsequence of A015733.

Programs

Mathematica

Select[Join[{1, 2}, Range[2, 420]^2], Divisible[EulerPhi[#] + 1, DivisorSigma[0, #]] &] (* Amiram Eldar, Mar 31 2021 *)

PARI

isA342665(n) = !((eulerphi(n)+1) % numdiv(n));