A056674 Number of squarefree divisors which are not unitary. Also number of unitary divisors which are not squarefree.

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 2, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 2, 1, 2, 0, 2, 0, 2, 0, 2, 0, 0, 0, 4, 0, 0, 2, 1, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 2, 2, 0, 0, 0, 2, 1, 0, 0, 4, 0, 0, 0, 2, 0, 4, 0, 2, 0, 0, 0, 2, 0, 2, 2, 3, 0, 0, 0, 2, 0

Offset: 1

Author

Labos Elemer, Aug 10 2000

Keywords

• nonn
• easy

Comments

Numbers of unitary and of squarefree divisors are identical, although the 2 sets are usually different, so sizes of parts outside overlap are also equal to each other.

Examples

			For n = 252, it has 18 divisors, 8 are unitary, 8 are squarefree, 2 belong to both classes, so 6 are squarefree but not unitary, thus a(252) = 6. The set {2,3,14,21,42} forms squarefree but non-unitary while the set {4,9,36,28,63,252} of same size gives the set of not squarefree but unitary divisors.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A000005, A007913, A034444, A055229, A055231, A056671.

Programs

Mathematica

Table[DivisorSum[n, 1 &, And[SquareFreeQ@ #, ! CoprimeQ[#, n/#]] &], {n, 105}] (* Michael De Vlieger, Jul 19 2017 *)
f[p_, e_] := If[e == 1, 2, 1]; a[1] = 0; a[n_] := 2^Length[fct = FactorInteger[n]] - Times @@ (f @@@ fct); Array[a, 100] (* Amiram Eldar, Jul 24 2024 *)

PARI

A034444(n) = (2^omega(n));
A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); } \\ Charles R Greathouse IV, Aug 13 2013
A055231(n) = n/A057521(n);
A056674(n) = (A034444(n) - numdiv(A055231(n)));
\\ Or:
A055229(n) = { my(c=core(n)); gcd(c, n/c); }; \\ Charles R Greathouse IV, Nov 20 2012
A056674(n) = ((2^omega(n)) - numdiv(core(n)/A055229(n)));
\\ Antti Karttunen, Jul 19 2017

PARI

a(n) = {my(f = factor(n), e = f[,2]); 2^omega(f) - prod(i = 1, #e, if(e[i] == 1, 2, 1));} \\ Amiram Eldar, Jul 24 2024

Python

from sympy import gcd, primefactors, divisor_count
from sympy.ntheory.factor_ import core
def a055229(n):
    c=core(n)
    return gcd(c, n//c)
def a056674(n): return 2**len(primefactors(n)) - divisor_count(core(n)//a055229(n))
print([a056674(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 19 2017

Formula

a(n) = A034444(n) - A056671(n) = A034444(n) - A000005(A055231(n)) = A034444(n) - A000005(A007913(n)/A055229(n)).