A244416 6-adic value of 1/n for n >= 1.

1, 6, 6, 36, 1, 6, 1, 216, 36, 6, 1, 36, 1, 6, 6, 1296, 1, 36, 1, 36, 6, 6, 1, 216, 1, 6, 216, 36, 1, 6, 1, 7776, 6, 6, 1, 36, 1, 6, 6, 216, 1, 6, 1, 36, 36, 6, 1, 1296, 1, 6, 6, 36, 1, 216, 1, 216, 6, 6, 1, 36, 1, 6, 36, 46656, 1, 6, 1, 36, 6, 6, 1, 216, 1, 6, 6, 36, 1, 6, 1, 1296, 1296, 6, 1, 36, 1, 6

Offset: 1

Author

Wolfdieter Lang, Jun 30 2014

Keywords

• nonn
• easy

Comments

For the definition of 'g-adic value of x', called |x|_g with g an integer >= 2, see the Mahler reference, p. 7. Sometimes also called g-adic absolute value of x. If g is not a prime then this is called a non-archimeden pseudo-valuation. See Mahler, p. 10.

Examples

			a(6) = 6^max(1,1) = 6^1 = 6. a(12) = 6^max(2,1) = 6^2 = 36,
a(18) = 6^max(1,2) = 36, a(24) = 6^max(3,1) = 6^3 = 216, ...
a(2) = 6^1 = 6, a(8) = 6^3 = 216, a(14) = 6^1 = 6, ...
a(3) = 6^1 = 6, a(9) = 6^2 = 36, a(15) = 6^1 = 6, ...
a(4) = 6^2 = 36, a(10) = 6^1 = 6, a(16) = 6^4 = 1296, ...
		

References

• Kurt Mahler, p-adic numbers and their functions, second ed., Cambridge University Press, 1981.

Crossrefs

Cf. A244417, A006519 (g=2), A038500 (g=3), A240226 (g=4), A060904 (g=5).

Programs

Mathematica

a[n_] := 6^Max[IntegerExponent[n, {2, 3}]]; Array[a, 100] (* Amiram Eldar, Aug 19 2024 *)

PARI

a(n) = 6^max(valuation(n, 2), valuation(n, 3)); \\ Amiram Eldar, Aug 19 2024

Formula

a(n) = 1 if n == 1 or 5 (mod 6). a(n) = 6^max(A007814(n), A007949(n)) if n == 0 (mod 6), a(n) = 6^A007814(n) if n == 2 or 4 (mod 6), a(n) = 6^A007949(n) if n == 3 (mod 6). The exponents, called f(1/n) in the Mahler reference, are given in A244417(n).

a(n) = 6^A244417(n). - Amiram Eldar, Aug 19 2024