A269299 Maximum number of times a nine-digit number with non-repeating, nonzero digits can be divided by the n-th prime until a non-integer result is returned.

16, 13, 7, 7, 5, 5, 5, 4, 4, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset

1,1

Comments

Sequence was constructed with a simple game in mind, one done by hand or by calculator, where one takes a nine-digit number with non-repeating nonzero digits and divides it by three until a non-integer result is returned. The potential best play is then of interest for this game, or the maximum number of times three divides such a number. The sequence extends this idea to all primes under the same rules and outputs the maximum number of times the n-th prime divides a number in this form.

The largest such prime that would give a nonzero solution is the 6289143rd prime, namely 109739359.

Links

Table of n, a(n) for n=1..96.

Example

For n=2, 3 divides 618597324 13 times, and this is the highest achievable valuation. In comparison, if the rules regarding nonzero and non-repeating were lifted, the number 3^18 = 387420489 would be allowed, and a(2) would be 18.

Maple

with(LinearAlgebra)*with(combinat)*with(numtheory):
A := permute(9):
B := 0:
for d to 6289143 do
a[d] := 0:
end do:
for i to factorial(9) do
B := ifactors(Determinant(Multiply(`<|>`(`<, >`(100000000), `<, >`(10000000), `<, >`(1000000), `<, >`(100000), `<, >`(10000), `<, >`(1000), `<, >`(100), `<, >`(10), `<, >`(1)), Transpose(convert(A[i], Matrix))))):
C := nops([op(B[2])]):
for j to C do
if a[pi(B[2, j, 1])] < B[2, j, 2] then
a[pi(B[2, j, 1])] := B[2, j, 2]:
end if:
end do:
end do:
seq(a[i], i = 1 .. 500);

Mathematica

s = FromDigits /@ Permutations@ Range@ 9; Table[Max@ Map[Length@ NestWhileList[#/Prime@ n &, #, IntegerQ] &, s], {n, 40}] - 2 (* Michael De Vlieger, Feb 23 2016 *)

Prog

(PARI) a(n) = {my(p=prime(n)); my(mv = 0); for (i=0, 9!, mv = max(valuation(subst(Pol(numtoperm(9, i)), x, 10), p), mv); ); mv; } \\ Michel Marcus, Feb 26 2016

Crossrefs

Sequence in context: A291426 A299524 A070534 * A070577 A184752 A291468

Adjacent sequences: A269296 A269297 A269298 * A269300 A269301 A269302

Keyword

nonn,base

Author

Aaron Benham, Feb 21 2016

Status

approved