

A056674


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


3



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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

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. Set {2,3,14,21,42} forms squarefree but nonunitary while 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 *)


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
(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 map(a056674, range(1, 101)) # Indranil Ghosh, Jul 19 2017


CROSSREFS

Cf. A000005, A007913, A034444, A000005, A055231, A055229, A056671.
Sequence in context: A070138 A024153 A079127 * A227761 A037188 A276847
Adjacent sequences: A056671 A056672 A056673 * A056675 A056676 A056677


KEYWORD

nonn,changed


AUTHOR

Labos Elemer, Aug 10 2000


STATUS

approved



