A054008 -id:A054008 - cp's OEIS frontend

A343242 Nonprime numbers k such that A054008(k) = A054024(k).

1, 20, 196, 368, 650, 672, 780, 836, 1888, 2352, 3192, 11096, 17816, 20496, 30240, 51060, 84660, 130304, 979992, 1848964, 2291936, 3767100, 4526272, 8353792, 15126992, 15287976, 23569920, 33468416, 45532800, 74899952, 381236216, 623799776, 712023296, 1845991216

Offset: 1

Amiram Eldar, Apr 08 2021

nonn

If p is an odd prime then A054008(p) = A054024(p) = 2. Therefore, this sequence is restricted to nonprime numbers.

			20 is a term since it is a nonprime number and A054008(20) = A054024(20) = 2.

Cf. A000005, A000203, A054008, A054024.

Subsequences: A047728, A245782.

Select[Range[10^6], !PrimeQ[#] && Mod[#, DivisorSigma[0, #]] == Mod[DivisorSigma[1, #], #] &]

A033950 Refactorable numbers: number of divisors of k divides k. Also known as tau numbers.

Original entry on oeis.org

1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, 104, 108, 128, 132, 136, 152, 156, 180, 184, 204, 225, 228, 232, 240, 248, 252, 276, 288, 296, 328, 344, 348, 360, 372, 376, 384, 396, 424, 441, 444, 448, 450, 468, 472, 480, 488, 492, 504, 516, 536

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk)

Kennedy and Cooper show that this sequence has density zero.
Spiro showed more precisely that the number of refactorable numbers less than x is asymptotic to (x/sqrt(log x))(log(log x))^(-1+o(1)). - David Eppstein, Aug 25 2014
Numbers k such that the equation gcd(k,x) = tau(k) has solutions. - Benoit Cloitre, Jun 10 2002
Refactorable numbers are the fixed points of A009230. - Labos Elemer, Nov 18 2002
Let ref(n) denote the characteristic function of the refactorable numbers. Then ref(n) = 1 + floor(n/d(n)) - ceiling(n/d(n)), where d(n) is the number of divisors of n. - Wesley Ivan Hurt, Jan 09 2013, Feb 15 2013
An odd number with an even number of divisors cannot be in the sequence by definition. Therefore all odd terms are squares (A000290). - Ivan N. Ianakiev, Aug 25 2013
A054008(k) = k mod A000005(k). - Reinhard Zumkeller, Sep 17 2014
The only squarefree terms are 1 and 2: if x is a squarefree number that is a product of n distinct primes, its number of divisors is 2^n, so x is refactorable if it contains 2^n as a factor, but that makes it nonsquarefree unless n = 0, 1, hence x = 1, 2. - Waldemar Puszkarz, Jun 10 2016
Every positive integer k occurs as tau(m) for some m in the sequence. If the factorization of k is Product p_i^e_i, then Product p_i^(p_i^e_i-1) has the specified property. For k prime, this is the only such number. - Franklin T. Adams-Watters, Jan 14 2017
Zelinsky (2002) proved that for any j > 0 and for sufficiently large m the number of terms not exceeding m is > j*pi(m), where pi(m) = A000720(m). - Amiram Eldar, Feb 20 2021
Numbers m such that the ratio (number of non-divisors of m)/(number of divisors of m) = A049820(m)/A000005(m) is an integer. - Michel Lagneau, Apr 04 2025

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B12, pp. 102-103.
New Scientist, Sep 05 1998, p. 17, para. 3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Kushagr Ahuja, Patrick Lei and Dylan Pentland, Tau ideals in number fields, PROMYS 2017.
Alan Bundy, Simon Colton and Toby Walsh, HR - A system for Machine Discovery in Finite Algebras, ECAI 1998.
Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2 (1999), Article 99.1.2.
Simon Colton, HR - Automatic Theory Formation in Pure Mathematics.
Robert E. Kennedy and Curtis N. Cooper, Tau numbers, natural density and Hardy and Wright's Theorem 437, International Journal of Mathematics and Mathematical Sciences, Vol. 13, No. 2 (1990), pp. 383-386.
Claudia Spiro, How often is the number of divisors of n a divisor of n?, J. Number Theory, Vol. 21, No. 1 (1985), pp. 81-100.
Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results , Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.

Cf. A000005, A000720, A039819, A036762, A051278, A051279, A051280, A036763.
Cf. A235353 (subsequence).
Cf. A054008, A281188.

a033950 n = a033950_list !! (n-1)
a033950_list = [x | x <- [1..], x `mod` a000005 x == 0]
-- Reinhard Zumkeller, Dec 28 2011

[ n: n in [1..540] | n mod #Divisors(n) eq 0 ]; // Klaus Brockhaus, Apr 29 2009

with(numtheory):
A033950 := proc(n)
    option remember:
    local k:
    if n=1 then
        return 1:
    else
        for k from procname(n-1)+1 do
            if type(k/tau(k), integer) then
                return k:
            end if:
        end do:
    end if:
end proc:
seq(A033950(n), n=1..56); # Nathaniel Johnston, May 04 2011

Do[If[IntegerQ[n/DivisorSigma[0, n]], Print[n]], {n, 1, 1000}]
Select[ Range[559], Mod[ #, DivisorSigma[0, # ]] == 0 &]
Select[Range[550], Divisible[ #, DivisorSigma[0, # ]]&] (* Waldemar Puszkarz, Jun 10 2016 *)

isA033950(n)=n%numdiv(n)==0 \\ Charles R Greathouse IV, Jun 10 2011

from sympy import divisor_count
print([n for n in range(1, 1001) if not n % divisor_count(n)]) # Indranil Ghosh, May 03 2017

More terms from Erich Friedman

A336040 Characteristic function of refactorable numbers (A033950).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0

Offset: 1

Wesley Ivan Hurt, Jul 07 2020

			a(1) = 1 since d(1) = 1 and 1 divides 1.
a(2) = 1 since d(2) = 2 and 2 divides 2.
a(3) = 0 since d(3) = 2, but 2 does not divide 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Eric Weisstein's World of Mathematics, Refactorable Number
Index entries for characteristic functions

Cf. A000005, A033950, A054008, A336041 (inverse Möbius transform), A335182, A335665, A349322.

a[n_] := Boole @ Divisible[n, DivisorSigma[0, n]]; Array[a, 100] (* Amiram Eldar, Jul 08 2020 *)

a(n) = n%numdiv(n) == 0; \\ Michel Marcus, Jul 07 2020

a(n) = 1 - ceiling(n/d(n)) + floor(n/d(n)), where d(n) is the number of divisors of n (A000005).
a(n) = [A054008(n) == 0], where [ ] is the Iverson bracket. - Antti Karttunen, Nov 24 2021
a(p) = 0 for odd primes p. - Wesley Ivan Hurt, Nov 28 2021

A054009 n read modulo (number of proper divisors of n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 1, 1, 0, 2, 0, 2, 0, 0, 0, 3, 0, 0, 0, 1, 0, 3, 1, 2, 0, 3, 0, 2, 0, 2, 0, 1, 2, 4, 0, 2, 0, 5, 0, 0, 0, 4, 0, 1, 0, 3, 1, 0, 0, 2, 0, 5, 1, 0, 0, 1, 0, 5, 0, 2, 3, 4, 2, 3, 0, 3, 0, 0, 0, 6, 0, 2, 0, 1, 2, 1, 0, 8, 1, 1, 0, 7, 1, 2, 0, 4, 0, 2, 1, 2, 0, 1, 2, 8, 0, 3, 4, 4, 0, 4, 0, 6, 0, 1

Offset: 2

Asher Auel, Jan 12 2000

nonn

Cf. A000005, A032741, A054008.

Maple
```
[ seq( i mod (tau(i) - 1), i=2..150) ];
```

Table[Mod[n,DivisorSigma[0,n]-1],{n,2,110}] (* Harvey P. Dale, Dec 05 2015 *)

a(n) = n % (numdiv(n) - 1); \\ Michel Marcus, Nov 21 2019

from sympy import divisor_count
def A054009(n): return n%(divisor_count(n)-1) # Chai Wah Wu, Mar 14 2023

a(n) = n mod (tau(n) - 1), for n>1.

A368625 Characteristic function of non-refactorable numbers (A159973).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1

Offset: 1

Wesley Ivan Hurt, Jan 01 2024

Antti Karttunen, Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, Refactorable Number
Index entries for characteristic functions

Cf. A000005, A033950 (refactorable numbers), A054008, A159973, A336040.

Table[(Ceiling[n/DivisorSigma[0, n]] - Floor[n/DivisorSigma[0, n]]), {n, 100}]

A368625(n) = !!(n%numdiv(n)); \\ Antti Karttunen, Jan 17 2025

a(n) = ceiling(n/d(n))-floor(n/d(n)), where d(n) is the number of divisors of n (A000005).
a(n) = 1 - A336040(n).
a(n) = [A054008(n) > 0], where [ ] is the Iverson bracket. - Antti Karttunen, Jan 17 2025

Data section extended up to a(105) by Antti Karttunen, Jan 17 2025

cp's OEIS Frontend

A343242 Nonprime numbers k such that A054008(k) = A054024(k).

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A033950 Refactorable numbers: number of divisors of k divides k. Also known as tau numbers.

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A336040 Characteristic function of refactorable numbers (A033950).

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A054009 n read modulo (number of proper divisors of n).

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A368625 Characteristic function of non-refactorable numbers (A159973).

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