A000190 - OEIS

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

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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.