A080825 -id:A080825 - cp's OEIS frontend

A082399 a(1) = 1; thereafter, a(n) is the smallest nonnegative number such that the number Sum_{i=1..n} a(i)*10^(n-i) is divisible by n.

1, 0, 2, 0, 0, 0, 5, 6, 4, 0, 5, 10, 2, 2, 5, 6, 16, 8, 14, 0, 7, 18, 19, 2, 5, 10, 6, 0, 25, 20, 2, 20, 17, 12, 20, 28, 13, 4, 13, 30, 16, 20, 36, 4, 35, 28, 28, 16, 29, 10, 39, 14, 12, 4, 50, 20, 14, 24, 7, 50, 14, 54, 55, 18, 10, 44, 62, 52, 63, 50, 7, 18, 6, 62, 55, 54, 54, 54, 35, 10

Offset: 1

N. J. A. Sloane, Feb 08 2009

nonn

Suggested by studying A144688. If all a(n) had turned out to be in the range 0 to 9 then this sequence would have produced a counterexample to the assertion that A144688 is finite.

The old entry with this A-number was a duplicate of A080825.

			After we have the first 11 terms, 1,0,2,0,0,0,5,6,4,0,5, the next number x must be chosen so that 102000564050 + x is divisible by 12; this implies that x = 10.

Paul Tek, Table of n, a(n) for n = 1..10000

See A051883 for another version. Cf. A144688.

M:=80; a[1]:=1; N:=1;
for n from 2 to M do
N:=10*N; t2:=N mod n;
if t2 = 0 then a[n]:=0; else a[n]:=n-t2; fi;
N:=N+a[n]; od: [seq(a[n],n=1..M)];

A082404 Triangle in which n-th row gives trajectory of n under the map x -> x/2 if x is even, x -> x-1 if x is odd, stopping when we reach 1.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 2, 1, 5, 4, 2, 1, 6, 3, 2, 1, 7, 6, 3, 2, 1, 8, 4, 2, 1, 9, 8, 4, 2, 1, 10, 5, 4, 2, 1, 11, 10, 5, 4, 2, 1, 12, 6, 3, 2, 1, 13, 12, 6, 3, 2, 1, 14, 7, 6, 3, 2, 1, 15, 14, 7, 6, 3, 2, 1, 16, 8, 4, 2, 1, 17, 16, 8, 4, 2, 1

Offset: 1

Cino Hilliard, Apr 14 2003

If you write down 0 when dividing by 2, 1 when subtracting 1, the trajectory gives the binary expansion of n.

The length of the n-th row of the triangle is A056792(n). - Nathaniel Johnston, Apr 21 2011

			Triangle begins:
  1;
  2, 1;
  3, 2, 1;
  4, 2, 1,
  5, 4, 2, 1;
  6, 3, 2, 1;
  7, 6, 3, 2, 1;
  8, 4, 2, 1;
  9, 8, 4, 2, 1;
  ...

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A056792, A080825.

A082404 := proc(n,k) option remember: if(k = 1)then return n:elif(A082404(n,k-1) mod 2 = 0)then return A082404(n,k-1)/2: else return A082404(n,k-1)-1: fi: end:
for n from 1 to 20 do k:=1: while A082404(n,k)>=1 do printf("%d, ",A082404(n,k)); k:=k+1: od:printf("\n");od: # Nathaniel Johnston, Apr 21 2011

T(n, 1) = n, T(n, 2) = A029578(n).

More terms and changed offset from Nathaniel Johnston, Apr 21 2011

A081169 Triangle in which n-th row gives trajectory of n (including n itself) under the map x -> x/2 if x is even, x -> 3*x-1 if x is odd, stopping when reaching 1, 5 or 17.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 8, 4, 2, 1, 4, 2, 1, 5, 14, 7, 20, 10, 5, 6, 3, 8, 4, 2, 1, 7, 20, 10, 5, 8, 4, 2, 1, 9, 26, 13, 38, 19, 56, 28, 14, 7, 20, 10, 5, 10, 5, 11, 32, 16, 8, 4, 2, 1, 12, 6, 3, 8, 4, 2, 1, 13, 38, 19, 56, 28, 14, 7, 20, 10, 5, 14, 7, 20, 10, 5, 15, 44, 22, 11, 32, 16, 8, 4, 2, 1, 16

Offset: 1

Cino Hilliard, Apr 16 2003

It is conjectured that the sequence will always end in one of three loops: 1, 2,1,1, ...; 5 14 7 20 10 5...; or 17 50 25 74 37 110 55 164 82 41 122 61 182 91 272 136 68 34 17...

Cf. A080825.

xnm3(n) = { print1(1" "2" "1" "); for(x=2,n, x1=x; print1(x1" "); while(x1>1, if(x1%2==0,x1/=2,x1 = 3*p-1); print1(x1" "); if(x1==5 || x1==17,break); ) ) }

A081170 Triangle in which n-th row gives trajectory of n (including n itself) under the map x -> x/2 if x is even, x -> x-3 if x is odd, stopping when reach 0 or 1.

Original entry on oeis.org

3, 0, 4, 2, 1, 5, 2, 1, 6, 3, 0, 7, 4, 2, 1, 8, 4, 2, 1, 9, 6, 3, 0, 10, 5, 2, 1, 11, 8, 4, 2, 1, 12, 6, 3, 0, 13, 10, 5, 2, 1, 14, 7, 4, 2, 1, 15, 12, 6, 3, 0, 16, 8, 4, 2, 1, 17, 14, 7, 4, 2, 1, 18, 9, 6, 3, 0, 19, 16, 8, 4, 2, 1, 20, 10, 5, 2, 1, 21, 18, 9, 6, 3, 0, 22, 11, 8, 4, 2, 1, 23, 20, 10, 5, 2

Offset: 3

Cino Hilliard, Apr 16 2003

Cf. A080825.

xn1m3(n) = { for(x=3,n, x1=x; print1(x1" "); while(x1>1, if(x1%2==0,x1/=2,x1=x1-3); print1(x1" "); ) ) }

A081171 Triangle in which n-th row gives trajectory of n (including n itself) under the map x -> x/2 if x is even, x -> 3x-3 if x is odd, stopping when reaching one of 1, 3, 5, 15, 51.

Original entry on oeis.org

1, 0, 2, 1, 3, 0, 4, 2, 1, 5, 2, 1, 6, 3, 7, 4, 2, 1, 8, 4, 2, 1, 9, 6, 3, 10, 5, 11, 8, 4, 2, 1, 12, 6, 3, 13, 10, 5, 14, 7, 4, 2, 1, 15, 12, 6, 3, 16, 8, 4, 2, 1, 17, 14, 7, 4, 2, 1, 18, 9, 6, 3, 19, 16, 8, 4, 2, 1, 20, 10, 5, 21, 18, 9, 6, 3, 22, 11, 8, 4, 2, 1, 23, 20, 10, 5, 24, 12, 6, 3, 25, 22

Offset: 1

Cino Hilliard, Apr 16 2003

Cf. A080825.

xn1m3(n) = { for(x=1,n, x1=x; print1(x1" "); while(x1>1, if(x1%2==0,x1/=2,x1=3*x1-3); print1(x1" "); ) ) }

cp's OEIS Frontend

A082399 a(1) = 1; thereafter, a(n) is the smallest nonnegative number such that the number Sum_{i=1..n} a(i)*10^(n-i) is divisible by n.

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A082404 Triangle in which n-th row gives trajectory of n under the map x -> x/2 if x is even, x -> x-1 if x is odd, stopping when we reach 1.

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A081169 Triangle in which n-th row gives trajectory of n (including n itself) under the map x -> x/2 if x is even, x -> 3*x-1 if x is odd, stopping when reaching 1, 5 or 17.

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PARI

A081170 Triangle in which n-th row gives trajectory of n (including n itself) under the map x -> x/2 if x is even, x -> x-3 if x is odd, stopping when reach 0 or 1.

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PARI

A081171 Triangle in which n-th row gives trajectory of n (including n itself) under the map x -> x/2 if x is even, x -> 3x-3 if x is odd, stopping when reaching one of 1, 3, 5, 15, 51.

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PARI