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A

abs absboxchar absolute_real_time acos acosh acot acoth acsc acsch activate activecontexts adapt_depth addcol additive addmatrices addrow adjoin adjoint airy_ai airy_bi airy_dai airy_dbi algebraic algepsilon algexact algsys alias aliases allbut allroots alphabetic alphacharp alphanumericp and antisymmetric append appendfile apply apply1 apply2 applyb1 apropos args array arrayapply arrayinfo arraymake arrays ascii asec asech asin asinh askexp askinteger asksign assoc assoc_legendre_p assoc_legendre_q assume assume_external_byte_order assume_pos assume_pos_pred assumescalar at atan atan2 atanh atom atomgrad atvalue augcoefmatrix

B

backsubst backtrace bashindices batch batchload bc2 belln berlefact bern bernpoly bessel_i bessel_j bessel_k bessel_y besselexpand beta beta_args_sum_to_integer beta_expand beta_incomplete beta_incomplete_generalized beta_incomplete_regularized bezout bfallroots bffac bfloat bfloatp bfpsi bfpsi0 bftorat bftrunc bfzeta binomial block blockmatrixp bothcoef box boxchar break breakup bug_report build_info buildq

C

cabs cardinality carg cartesian_product cauchysum cbffac ceiling cequal cequalignore cf cfdisrep cfexpand cflength cgreaterp cgreaterpignore changevar chaosgame charat charfun charlist charp charpoly chebyshev_t chebyshev_u cholesky christof cint clear_rules clessp clesspignore close closefile coeff coefmatrix col collapse collectterms color colorbox columnop columnspace columnswap combine commutative comp2pui compare compfile compile compile_file complex concat conjugate cons constant constantp constituent cont2part content context contexts contour_plot contract copy copylist copymatrix cos cosh cot coth covdiff create_list csc csch ctranspose cunlisp current_let_rule_package

D

deactivate debugmode declare declare_translated default_let_rule_package defcon define define_variable defint defmatch defrule deftaylor del delete delta demo demoivre denom dependencies depends derivabbrev derivdegree derivlist derivsubst describe desolve determinant detout diag_matrix diagmatrix diff digitcharp dim direct disjoin disjointp disolate disp dispflag dispform dispfun display display2d display_format_internal disprule dispterms distrib distribute_over divide divisors divsum do doallmxops domain domxexpt domxmxops domxnctimes dontfactor doscmxops doscmxplus dot0nscsimp dot0simp dot1simp dotassoc dotconstrules dotdistrib dotexptsimp dotident dotproduct dotscrules dpart draw draw2d draw3d dscalar

E

echelon eigens_by_jacobi eigenvalues eigenvectors eighth einstein elapsed_real_time elapsed_run_time ele2comp ele2polynome ele2pui elem elementp eliminate elliptic_e elliptic_ec elliptic_eu elliptic_f elliptic_kc elliptic_pi ematrix emptyp endcons entermatrix entier equal equiv_classes erf erf_generalized erf_representation erfc erfflag erfi errcatch error error_size error_syms errormsg euler ev eval eval_string evenp every evflag evfun evolution evolution2d example exp expand expandwrt expandwrt_denom expandwrt_factored expintegral_chi expintegral_ci expintegral_e expintegral_e1 expintegral_ei expintegral_li expintegral_shi expintegral_si expintexpand expintrep explicit explose expon exponentialize expop expt exptdispflag exptisolate exptsubst extremal_subset ezgcd

F

facexpand facsum facsum_combine factcomb factlim factor factorfacsum factorflag factorial factorial_expand factorout factorsum facts false fasttimes feature featurep features fib fibtophi fifth file_output_append file_search file_search_demo file_search_lisp file_search_maxima file_type filename_merge files fillarray find_root find_root_abs find_root_error find_root_rel first fix flatten flength float float2bf floatnump floor for forget fortindent fortran fortspaces fourth fposition fpprec fpprintprec freeof freshline fresnel_c fresnel_s full_listify fullmap fullmapl fullratsimp fullsetify funcsolve function functions fundef funmake

G

H

halfangles hankel hankel_1 hankel_2 help hermite hessian hgfred hilbert_matrix hipow horner hypergeometric hypergeometric_representation

I

ibase ic1 ic2 ident identfor identity if ifactors ifs ilt imagpart inchar ind indices inf infeval infinity infix inflag infolists inpart inrt intanalysis integer_partitions integerp integrate integrate_use_rootsof integration_constant integration_constant_counter intersect intersection intervalp intfaclim intopois intosum inv_mod inverse_jacobi_cd inverse_jacobi_cn inverse_jacobi_cs inverse_jacobi_dc inverse_jacobi_dn inverse_jacobi_ds inverse_jacobi_nc inverse_jacobi_nd inverse_jacobi_ns inverse_jacobi_sc inverse_jacobi_sd inverse_jacobi_sn invert invert_by_lu is isolate isolate_wrt_times isqrt

J

jacobi jacobi_cd jacobi_cn jacobi_cs jacobi_dc jacobi_dn jacobi_ds jacobi_nc jacobi_nd jacobi_ns jacobi_p jacobi_sc jacobi_sd jacobi_sn jacobian join julia

K

keepfloat kill killcontext kostka kron_delta kronecker_product

L

labels laguerre lambda lambert_w laplace lassociative last lcharp ldefint ldisp ldisplay legendre_p legendre_q length let let_rule_packages letrat letrules letsimp lfreeof lgtreillis lhospitallim lhs li limit limsubst linear linechar linel linenum linsolve linsolve_params linsolvewarn lispdisp listarith listarray listconstvars listdummyvars listify listofvars listp lmax lmin lmxchar load loadfile loadprint local locate_matrix_entry log log_gamma logabs logarc logconcoeffp logcontract logexpand lognegint lognumer logsimp logx logy lopow lowercasep lpart lreduce lsum ltreillis lu_backsub lu_factor

M

m1pbranch macroexpand macroexpand1 macroexpansion macros mainvar make_array make_random_state make_transform makebox makefact makegamma makelist makeset mandelbrot manual_demo map mapatom maperror maplist mapprint mat_cond mat_fullunblocker mat_norm mat_trace mat_unblocker matchdeclare matchfix matrix matrix_element_add matrix_element_mult matrix_element_transpose matrix_size matrixmap matrixp max maxapplydepth maxapplyheight maxima_tempdir maxima_userdir maxnegex maxposex maxpsifracdenom maxpsifracnum maxpsinegint maxpsiposint maxtayorder maybe member min minf minfactorial minor mod mode_check_errorp mode_check_warnp mode_checkp mode_declare mode_identity modulus moebius mon2schur multi_elem multi_orbit multi_pui multinomial multinomial_coeff multiplicative multiplicities multsym multthru myoptions

N

nary negdistrib negsumdispflag newcontext newdet newline next_prime nextlayerfactor niceindices niceindicespref ninth noeval nofix nolabels nonnegintegerp nonscalar nonscalarp not notequal noun noundisp nounify nouns nroots nterms nthroot nticks nullity nullspace num num_distinct_partitions num_partitions numberp numer numerval numfactor nusum nzeta nzetai nzetar

O

P

packagefile pade palette parabolic_cylinder_d parametric parse_string part part2cont partfrac partition partition_set partpol partswitch permanent permut permutations pfeformat phi pi pickapart piece playback plog plot2d plot3d plot_options pochhammer point_type points poisdiff poisexpt poisint poislim poismap poisplus poissimp poissubst poistimes poistrim polarform polydecomp polygon polymod polynome2ele polynomialp polytocompanion posfun postfix power_mod powerdisp powerseries powerset pred prederror prefix prev_prime primep primep_number_of_tests print printf printfile printpois printprops prodrac product programmode prompt properties props propvars psexpand psi ptriangularize pui pui2comp pui2ele pui2polynome pui_direct puireduc put

Q

qput quad_qag quad_qagi quad_qags quad_qawc quad_qawf quad_qaws quit qunit quotient

R

radcan radexpand radsubstflag random random_normal random_permutation rank rassociative rat ratalgdenom ratcoef ratdenom ratdenomdivide ratdiff ratdisrep rateinstein ratepsilon ratexpand ratfac rational rationalize ratmx ratnumer ratnump ratp ratprint ratriemann ratsimp ratsimpexpons ratsubst ratvars ratweight ratweights ratweyl ratwtlvl read read_array read_binary_array read_binary_list read_binary_matrix read_hashed_array read_list read_matrix read_nested_list readline readonly realonly realpart realroots rearray rectform refcheck rem remainder remarray rembox remcon remfunction remlet remove remrule remvalue rename reset residue resolvante resolvante_alternee1 resolvante_bipartite resolvante_diedrale resolvante_klein resolvante_klein3 resolvante_produit_sym resolvante_unitaire resolvante_vierer rest resultant return reveal reverse rhs riemann rinvariant risch rk rmxchar romberg room rootsconmode rootscontract rootsepsilon round row rowop rowswap rreduce run_testsuite

S

save savedef savefactors scalarmatrixp scalarp scaled_bessel_i scaled_bessel_i0 scaled_bessel_i1 scanmap schur2comp sconcat scopy scsimp scurvature sdowncase sec sech second sequal sequalignore set_partitions set_plot_option set_random_state set_tex_environment set_tex_environment_default setcheck setcheckbreak setdifference setelmx setequalp setify setp setup_autoload setval seventh sexplode show showratvars showtime sign signum simp simplode simpsum sin sinh sinsert sinvertcase sixth slength smake smismatch solve solvedecomposes solveexplicit solvefactors solvenullwarn solveradcan solvetrigwarn some somrac sort sparse specint spherical_bessel_j spherical_bessel_y spherical_hankel1 spherical_hankel2 spherical_harmonic splice split sposition sprint sqfr sqrt sqrtdispflag sremove sremovefirst sreverse ssearch ssort sstatus ssubst ssubstfirst staircase stardisp status stirling1 stirling2 strim striml strimr string stringdisp stringout stringp struve_h struve_l sublis sublis_apply_lambda sublist sublist_indices submatrix subset subsetp subst substinpart substpart substring subvar subvarp sum sumcontract sumexpand sumsplitfact supcase supcontext symbolp symmdifference symmetric syntax

T

tab tan tanh taylor taylor_logexpand taylor_order_coefficients taylor_simplifier taylor_truncate_polynomials taylordepth taylorinfo taylorp taytorat tcl_output tcontract tellrat tellsimp tellsimpafter tenth testsuite_files tex tex1 texput third throw time timedate timer timer_devalue timer_info tldefint tlimit tlimswitch to_lisp todd_coxeter toeplitz tokens totaldisrep totient tpartpol tr_array_as_ref tr_bound_function_applyp tr_file_tty_messagesp tr_float_can_branch_complex tr_function_call_default tr_numer tr_optimize_max_loop tr_semicompile tr_state_vars tr_warn_bad_function_calls tr_warn_fexpr tr_warn_meval tr_warn_mode tr_warn_undeclared tr_warn_undefined_variable tr_warnings_get tr_windy trace trace_options transcompile translate translate_file transpose transrun tree_reduce treillis treinat triangularize trigexpand trigexpandplus trigexpandtimes triginverses trigrat trigreduce trigsign trigsimp true trunc ttyoff

U

ultraspherical und undefined undiff union unique unit_step unknown unless unorder unsum untellrat untimer untrace uppercasep use_fast_arrays

V

values vandermonde_matrix var vect_cross verbify verbose

W

weyl while with_stdout write_binary_data write_data writefile

X

xaxis xlabel xreduce xthru

Y

Z

zeroa zerob zerobern zeroequiv

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Moebius Calculator

Moebius

Function: moebius (<n>) Represents the Moebius function.

$moebius (1); moebius (2 * 3 * 5); moebius (11 * 17 * 29 * 31); moebius (2^32); moebius (n); moebius (n = 12); moebius ([11, 11 * 13, 11 * 13 * 15]); moebius (matrix ([11, 12], [13, 14])); moebius ({21, 22, 23, 24});$

When <n> is product of k distinct primes, moebius(<n>) simplifies to (-1)^k; when <n> = 1, it simplifies to 1; and it simplifies to 0 for all other positive integers.

moebius distributes over equations, lists, matrices, and sets.

Examples:

          (%i1) moebius (1);
          (%o1)                           1
          (%i2) moebius (2 * 3 * 5);
          (%o2)                          - 1
          (%i3) moebius (11 * 17 * 29 * 31);
          (%o3)                           1
          (%i4) moebius (2^32);
          (%o4)                           0
          (%i5) moebius (n);
          (%o5)                      moebius(n)
          (%i6) moebius (n = 12);
          (%o6)                    moebius(n) = 0
          (%i7) moebius ([11, 11 * 13, 11 * 13 * 15]);
          (%o7)                      [- 1, 1, 1]
          (%i8) moebius (matrix ([11, 12], [13, 14]));
                                     [ - 1  0 ]
          (%o8)                      [        ]
                                     [ - 1  1 ]
          (%i9) moebius ({21, 22, 23, 24});
          (%o9)                      {- 1, 0, 1}

(%o1)                                true
(%i2)

Moebius Example

Related Examples

moebius-nusum

nusum(moebius(i),i,1,5);

moebius-nusum

nusum(moebius(i),i,1,...

moebius

moebius(5);

moebius

? moebius;

moebius-nusum-true

nusum(moebius(i),i,1,...

moebius-nusum-true

nusum(moebius(i),i,1,...

moebius-sum

sum (moebius(n),n,1,5);

moebius-nusum

nusum(moebius(i),i,1,...

moebius-nusum

nusum(moebius(i),i,1,...

moebius-nusum

nusum(moebius(i),i,1,5);