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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 non-unitary 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: A024153 A341523 A079127 * A336107 A227761 A037188

Adjacent sequences:  A056671 A056672 A056673 * A056675 A056676 A056677

KEYWORD

nonn

AUTHOR

Labos Elemer, Aug 10 2000

STATUS

approved

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Last modified March 7 19:13 EST 2021. Contains 341928 sequences. (Running on oeis4.)