A054009 -id:A054009 - cp's OEIS frontend

A054008 n read modulo (number of divisors of n).

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

Offset: 1

Asher Auel, Jan 12 2000

nonn

a(n)=0 iff n is a refactorable number (cf. A033950). - Franz Vrabec, Oct 16 2005

a(A066708(n)) = n and a(m) < n for m < A066708(n). - Reinhard Zumkeller, Sep 17 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A054009, A033950, A066708.

a054008 n = n `mod` a000005 n  -- Reinhard Zumkeller, Sep 17 2014

Maple
```
[ seq( i mod tau(i), i=1..130) ];
```

a[n_] := Mod[n, DivisorSigma[0, n]]; Array[a, 105] (* Jean-François Alcover, Sep 19 2017 *)

a(n) = n % numdiv(n); \\ Michel Marcus, Sep 19 2017

from sympy import divisor_count
def A054008(n): return n%divisor_count(n) # Chai Wah Wu, Mar 14 2023

a(n) = n mod tau(n).

A054010 Numbers n with property that n is divisible by the number of its proper divisors.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 11, 13, 15, 16, 17, 19, 20, 21, 23, 27, 29, 31, 33, 37, 39, 41, 42, 43, 45, 47, 50, 51, 53, 56, 57, 59, 61, 67, 69, 70, 71, 73, 75, 79, 83, 87, 89, 93, 97, 101, 103, 105, 107, 109, 111, 113, 120, 123, 127, 129, 131, 132, 137, 139, 141, 149, 151, 154

Offset: 1

Asher Auel, Jan 12 2000

nonn

All primes are in this sequence, having only one proper divisor. The specifically nonprime members of this sequence are in A055719. - Carl R. White, Jul 11 2012

			There are three proper divisors of 6, {1, 2, 3}, 6 is divisible by 3.

Carl R. White, Table of n, a(n) for n = 1..10000

Cf. A032741, A033950, A054009, A055719.

[seq(`if`(i mod (tau(i)-1) = 0,i,print( )), i=2..190)];

Select[Range[2, 100], IntegerQ[ #/(-1+DivisorSigma[0, # ])]&] (* Wouter Meeussen, Jun 07 2005 *)

Numbers n such that A054009(n) = 0.

A054011 n is not divisible by the number of its proper divisors.

Original entry on oeis.org

8, 9, 10, 12, 14, 18, 22, 24, 25, 26, 28, 30, 32, 34, 35, 36, 38, 40, 44, 46, 48, 49, 52, 54, 55, 58, 60, 62, 63, 64, 65, 66, 68, 72, 74, 76, 77, 78, 80, 81, 82, 84, 85, 86, 88, 90, 91, 92, 94, 95, 96, 98, 99, 100, 102, 104, 106, 108, 110, 112, 114, 115, 116, 117, 118, 119

Offset: 1

Asher Auel, Jan 17 2000

nonn

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

Cf. A000005, A032741, A054009, A054012.

with(numtheory): [seq(`if`(i mod (tau(i)-1) <> 0,i,print( )), i=2..190)];

Select[Range[2, 120], ! Divisible[#, DivisorSigma[0, #] - 1] &] (* Amiram Eldar, Aug 28 2019 *)

A054012 Nonzero values of n read modulo (number of proper divisors of n).

Original entry on oeis.org

2, 1, 1, 2, 2, 3, 1, 3, 1, 2, 3, 2, 2, 1, 2, 4, 2, 5, 4, 1, 3, 1, 2, 5, 1, 1, 5, 2, 3, 4, 2, 3, 3, 6, 2, 1, 2, 1, 8, 1, 1, 7, 1, 2, 4, 2, 1, 2, 1, 2, 8, 3, 4, 4, 4, 6, 1, 9, 5, 4, 2, 1, 1, 2, 1, 2, 1, 2, 4, 2, 5, 2, 4, 1, 2, 2, 3, 5, 8, 1, 2, 4, 1, 2, 2, 3, 7, 5, 3, 2, 2, 2, 6, 2, 4, 4, 1, 3, 1, 2, 1, 2, 6, 5, 1

Offset: 1

Asher Auel, Jan 17 2000

nonn

Cf. A000005, A032741, A054009, A054011.

with(numtheory):
[seq(`if`(i mod (tau(i)-1) <> 0,i mod (tau(i)-1),print( )), i=2..220)];

Cases[Table[Mod[n,DivisorSigma[0,n]-1],{n,2,200}],Except[0]] (* Harvey P. Dale, Nov 12 2011 *)

a(n) = A054011(A054009(n)).

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A054008 n read modulo (number of divisors of n).

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A054010 Numbers n with property that n is divisible by the number of its proper divisors.

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A054011 n is not divisible by the number of its proper divisors.

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A054012 Nonzero values of n read modulo (number of proper divisors of n).

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