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A020491 Numbers k such that sigma_0(k) divides phi(k). 11
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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)

MAPLE

with(numtheory);

A020491:=proc(q)

local n; for n from 1 to q do if (phi(n) mod tau(n))=0 then print(n); fi;

od; end:

A020491(1000000); # Paolo P. Lava, Jan 31 2013

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A000005, A000010.

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

Cf. A058891, A076688, A289585.

Sequence in context: A141114 A136443 A247459 * A168501 A173186 A047746

Adjacent sequences:  A020488 A020489 A020490 * A020492 A020493 A020494

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved

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Last modified January 15 21:50 EST 2021. Contains 340195 sequences. (Running on oeis4.)