A141115 -id:A141115

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A141113

Positive integers k such that d(d(k)) divides k, where d(k) is the number of divisors of k.

+10
4

1, 2, 4, 6, 12, 15, 16, 20, 21, 24, 27, 28, 32, 33, 36, 39, 40, 44, 48, 51, 52, 56, 57, 60, 64, 68, 69, 72, 76, 80, 84, 87, 88, 90, 92, 93, 96, 104, 108, 111, 112, 116, 120, 123, 124, 126, 128, 129, 132, 136, 141, 144, 148, 150, 152, 156, 159, 164, 172, 176, 177, 180

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Jinyuan Wang, Table of n, a(n) for n = 1..10000

EXAMPLE

28 has 6 divisors and 6 has 4 divisors. 4 divides 28, so 28 is in the sequence.

MAPLE

with(numtheory): a:=proc(n) if `mod`(n, tau(tau(n))) = 0 then n else end if end proc: seq(a(n), n=1..200); # Emeric Deutsch, Jun 05 2008

MATHEMATICA

Select[Range[200], Divisible[#, DivisorSigma[0, DivisorSigma[0, #]]]&] (* Harvey P. Dale, Feb 05 2012 *)

PROG

(PARI) is(k) = k%numdiv(numdiv(k)) == 0; \\ Jinyuan Wang, Feb 19 2019

CROSSREFS

Cf. A000005 (tau), A010553, A141114, A141115.

KEYWORD

nonn

AUTHOR

Leroy Quet, Jun 04 2008

EXTENSIONS

More terms from Emeric Deutsch, Jun 05 2008

STATUS

approved

A141114

Positive integers k where d(d(k)) is coprime to k, where d(k) is the number of divisors of k.

+10
3

1, 3, 5, 7, 8, 9, 10, 11, 13, 14, 17, 19, 22, 23, 25, 26, 29, 31, 34, 35, 37, 38, 41, 43, 45, 46, 47, 49, 53, 55, 58, 59, 61, 62, 63, 65, 67, 71, 73, 74, 75, 77, 79, 81, 82, 83, 85, 86, 89, 91, 94, 95, 97, 99, 100, 101, 103, 105, 106, 107, 109, 113, 115, 117, 118, 119, 121

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Includes all primes, squares of odd primes, and squarefree semiprimes coprime to 3. - Robert Israel, Dec 16 2019

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

EXAMPLE

26 has 4 divisors and 4 has 3 divisors. 3 is coprime to 26, so 26 is in the sequence.

MAPLE

filter:= proc(n) uses numtheory;

igcd(tau(tau(n)), n) = 1

end proc:

select(filter, [$1..200]); # Robert Israel, Dec 16 2019

MATHEMATICA

Select[Range[200], GCD[DivisorSigma[0, DivisorSigma[0, # ]], # ] == 1 &] (* Stefan Steinerberger, Jun 05 2008 *)

PROG

(Magma) [k:k in [1..130]|Gcd(k, #Divisors(#Divisors(k))) eq 1]; // Marius A. Burtea, Dec 16 2019

(PARI) is(n) = gcd(numdiv(numdiv(n)), n)==1 \\ Felix Fröhlich, Dec 16 2019