A020491 Numbers k such that sigma_0(k) divides phi(k).

1, 3, 5, 7, 8, 9, 10, 11, 13, 15, 17, 18, 19, 21, 23, 24, 26, 28, 29, 30, 31, 33, 34, 35, 37, 39, 40, 41, 43, 45, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 63, 65, 67, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 82, 83, 84, 85, 87, 88, 89, 90, 91, 93, 95, 97, 98, 99, 101, 102, 103, 104

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

In other words, numbers k such that d(k) divides phi(k).

From Enrique Pérez Herrero, Aug 11 2010: (Start)

sigma_0(k) divides phi(k) when:

k is an odd prime: A065091;

k is an odd squarefree number: A056911;

k = 2^m, where m <> 1 is a Mersenne number (A000225).

If d divides (p-1), with p prime, then p^(d-1) is in this sequence, as are p^(p-1), p^(p-2) and p^(-1+p^n).

(End)

phi(n) and d(n) are multiplicative functions, so if m and n are coprime and both of them are in this sequence then m*n is also in this sequence. - Enrique Pérez Herrero, Sep 05 2010

From Bernard Schott, Aug 14 2020: (Start)

The corresponding quotients are in A289585.

About the 3rd case of Enrique Pérez Herrero's comment: if k = 2^M_m, where M_m = 2^m - 1 is a Mersenne number >= 3 (A000225), then the corresponding quotient phi(k)/d(k) is the integer 2^(2^m-m-2) = A076688(m); hence, these numbers k, A058891 \ {2}, form a subsequence. (End)

Links

• Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
• Psychedelic Geometry Blogspot, Fermat and Mersenne Numbers Conjecture-(2)

Crossrefs

Cf. A000005, A000010.

Complement of A015733. [Enrique Pérez Herrero, Aug 11 2010]

Cf. A058891, A076688, A289585.

Programs

Mathematica

Select[ Range[ 105 ], IntegerQ[ EulerPhi[ # ]/DivisorSigma[ 0, # ] ]& ]

PARI

isok(k) = !(eulerphi(k) % numdiv(k)); \\ Michel Marcus, Aug 10 2020