A141113 -id:A141113 - cp's OEIS frontend

A141114 Positive integers k where d(d(k)) is coprime to k, where d(k) is the number of divisors of k.

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

Offset: 1

Leroy Quet, Jun 04 2008

nonn

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

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

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

Cf. A010553, A141113, A141115.

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

filter:= proc(n) uses numtheory;
  igcd(tau(tau(n)), n) = 1
end proc:
select(filter, [$1..200]); # Robert Israel, Dec 16 2019

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

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

More terms from Stefan Steinerberger, Jun 05 2008

A141115 Those positive integers k where both d(d(k)) is not coprime to k and d(d(k)) does not divide k, where d(k) is the number of divisors of k.

Original entry on oeis.org

18, 30, 42, 50, 54, 66, 70, 78, 98, 102, 110, 114, 130, 138, 140, 154, 160, 162, 170, 174, 182, 186, 190, 200, 220, 222, 224, 230, 238, 242, 246, 250, 258, 260, 266, 282, 286, 290, 308, 310, 315, 318, 322, 338, 340, 350, 352, 354, 364, 366, 370, 374, 380, 392

Offset: 1

Leroy Quet, Jun 04 2008

nonn

			50 has 6 divisors and 6 has 4 divisors. 4 is not coprime to 50 and 4 does not divide 50. So 50 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A010553, A141113, A141114.

Select[Range[400], GCD[DivisorSigma[0,DivisorSigma[0, # ]], # ] > 1 && Mod[ #, DivisorSigma[0, DivisorSigma[0, # ]]] > 0 &] (* Stefan Steinerberger, Jun 05 2008 *)

More terms from Stefan Steinerberger, Jun 05 2008

A174457 Infinitely refactorable numbers: numbers k such that each iteration under the map x -> A000005(x) produces a divisor of k.

Original entry on oeis.org

1, 2, 12, 24, 36, 60, 72, 84, 96, 108, 132, 156, 180, 204, 228, 240, 252, 276, 288, 348, 360, 372, 396, 444, 468, 480, 492, 504, 516, 564, 600, 612, 636, 640, 672, 684, 708, 720, 732, 792, 804, 828, 852, 864, 876, 936, 948, 972, 996, 1044, 1056, 1068, 1116, 1152

Offset: 1

Matthew Vandermast, Dec 04 2010

nonn

In other words, let d^1(n) = A000005(n) and, for all positive integers k, let d^(k+1)(n) = A000005(d^k(n)). Sequence lists numbers n with the property that every such value of d^k(n) divides n.
A141586 is a subsequence. Is A110821 a subsequence?
Not a subsequence of A141551: 504 is the smallest term in this sequence not member of A141551.
a(n) is even for all n, since for any n >= 2, d^k(n) = 2 for some k. Proof: {d^k(n)} is a nonincreasing sequence of k, so it must stablize at a fixed point of the map x -> A000005(x), namely x = 1 or 2. But d^k(n) = 1 for some k implies that n = 1. - Jianing Song, Apr 20 2022

			9 has 3 divisors, and 9 is a multiple of 3. But 3 has 2 divisors, and 9 is not a multiple of 2. Hence, 9 does not belong to this sequence.
36 has 9 divisors, 9 has 3 divisors, 3 has 2 divisors, and 9, 3, and 2 are all divisors of 36. (Since 2 has 2 divisors, all further steps produce a value of 2.) Hence, 36 belongs to this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A036459 (number of steps of the map), A000005 (d(n): number of divisors).
Cf. A010553 (d(d(n))), A036450 (d^3(n)), A036452 (d^4(n)), A036453 (d^5(n)).
Subsequence of A033950 (refactorable numbers: d(n) | n) and A141113 (d(d(n))| n).

is_A174457(n, d=n)=!until(d<3, n%(d=numdiv(d)) && return) \\ M. F. Hasler, Dec 05 2010, updated PARI syntax Apr 16 2022

Edited by M. F. Hasler, Apr 16 2022

cp's OEIS Frontend

A141114 Positive integers k where d(d(k)) is coprime to k, where d(k) is the number of divisors of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Extensions

A141115 Those positive integers k where both d(d(k)) is not coprime to k and d(d(k)) does not divide k, where d(k) is the number of divisors of k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A174457 Infinitely refactorable numbers: numbers k such that each iteration under the map x -> A000005(x) produces a divisor of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions