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

1,12

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.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from exponents in factorization of n.

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A354527 A341523 A079127 * A349798 A336107 A350251

Adjacent sequences: A056671 A056672 A056673 * A056675 A056676 A056677

Keyword

nonn,easy

Author

Labos Elemer, Aug 10 2000

Status

approved