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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 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: 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

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Last modified December 8 04:31 EST 2019. Contains 329850 sequences. (Running on oeis4.)