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A000190 Number of solutions to x^4 == 0 (mod n).

%I A000190 #39 Dec 18 2023 01:45:57
%S A000190 1,1,1,2,1,1,1,4,3,1,1,2,1,1,1,8,1,3,1,2,1,1,1,4,5,1,9,2,1,1,1,8,1,1,
%T A000190 1,6,1,1,1,4,1,1,1,2,3,1,1,8,7,5,1,2,1,9,1,4,1,1,1,2,1,1,3,16,1,1,1,2,
%U A000190 1,1,1,12,1,1,5,2,1,1,1,8,27,1,1,2,1,1,1,4,1,3
%N A000190 Number of solutions to x^4 == 0 (mod n).
%C A000190 Shadow transform of fourth powers A000583. - _Michel Marcus_, Jun 06 2013
%H A000190 T. D. Noe, <a href="/A000190/b000190.txt">Table of n, a(n) for n = 1..10000</a>
%H A000190 Henry Bottomley, <a href="http://fs.gallup.unm.edu/Bottomley-Sm-Mult-Functions.htm">Some Smarandache-type multiplicative sequences</a>.
%H A000190 Lorenz Halbeisen, <a href="http://www.iam.fmph.uniba.sk/amuc/_vol-74/_no_2/_halbeisen/halbeisen.html">A number-theoretic conjecture and its implication for set theory</a>, Acta Math. Univ. Comenianae 74(2) (2005), 243-254.
%H A000190 Lorenz Halbeisen and Norbert Hungerbuehler, <a href="http://math.berkeley.edu/~halbeis/publications/pdf/seq.pdf">Number theoretic aspects of a combinatorial function</a>, Notes on Number Theory and Discrete Mathematics 5 (1999), 138-150.
%H A000190 OEIS Wiki, <a href="https://oeis.org/wiki/Shadow_transform">Shadow transform</a>.
%H A000190 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%F A000190 Multiplicative with a(p^e) = p^[3e/4]. - _David W. Wilson_, Aug 01 2001
%F A000190 Dirichlet g.f.: zeta(4*s-3) * Product_{p prime} (1 + 1/p^s + 1/p^(2*s-1) + 1/p^(3*s-2)). - _Amiram Eldar_, Dec 18 2023
%t A000190 Array[ Function[ n, Count[ Array[ PowerMod[ #, 4, n ]&, n, 0 ], 0 ] ], 100 ]
%t A000190 f[p_, e_] := p^Floor[3*e/4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 19 2020 *)
%o A000190 (PARI) a(n)=my(f=factor(n));prod(i=1,#f[,1],f[i,1]^(3*f[i,2]\4)) \\ _Charles R Greathouse IV_, Jun 07 2013
%Y A000190 Cf. A000583.
%K A000190 nonn,mult,easy
%O A000190 1,4
%A A000190 _N. J. A. Sloane_