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

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, 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, 1, 1, 1, 12, 1, 1, 5, 2, 1, 1, 1, 8, 27, 1, 1, 2, 1, 1, 1, 4, 1, 3

Offset

1,4

Comments

Shadow transform of fourth powers A000583. - Michel Marcus, Jun 06 2013

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Henry Bottomley, Some Smarandache-type multiplicative sequences.

Lorenz Halbeisen, A number-theoretic conjecture and its implication for set theory, Acta Math. Univ. Comenianae 74(2) (2005), 243-254.

Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5 (1999), 138-150.

OEIS Wiki, Shadow transform.

N. J. A. Sloane, Transforms.

Formula

Multiplicative with a(p^e) = p^[3e/4]. - David W. Wilson, Aug 01 2001

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

Mathematica

Array[ Function[ n, Count[ Array[ PowerMod[ #, 4, n ]&, n, 0 ], 0 ] ], 100 ]
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 *)

Prog

(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

Crossrefs

Cf. A000583.

Sequence in context: A104445 A359762 A000189 * A348037 A003557 A073752

Adjacent sequences: A000187 A000188 A000189 * A000191 A000192 A000193

Keyword

nonn,mult,easy

Author

N. J. A. Sloane

Status

approved