A290223 Algorithm: s(k) = n. s(k+1) = s(k) - digitsum(s(k))^2 if s(k) >= 0 and s(k+1) = s(k)+digitsum(abs(s(k)))^2 if s(k) < 0. Below gives the end behavior for each number n.

0, 2, 3, 6, 6, 6, 3, 11, 9, 9, 3, 3, 6, 6, 6, 3, 11, 9, 0, 3, 3, 6, 2, 6, 3, 11, 9, 9, 11, 3, 6, 3, 6, 3, 11, 9, 9, 11, 3, 6, 3, 6, 3, 6, 9, 9, 11, 3, 6, 3, 6, 3, 6, 9, 9, 11, 3, 6, 6, 6, 3, 2, 9, 9, 11, 3, 6, 6, 6, 3, 3, 9, 9, 11, 3, 6, 2, 6, 3, 3, 0, 9, 11, 3, 6, 6, 6, 3, 6, 9, 9, 11, 3, 6, 6, 6, 3, 6, 9, 9, 3, 3, 6, 3, 6, 3, 3, 9, 9

Offset: 1

Author

Derek Orr, Jul 24 2017

Keywords

• nonn
• base

Comments

0 means the sequence s(k) becomes the 0 sequence.

2 means the sequence s(k) becomes 2, -2, 2, -2, ...

3 means the sequence s(k) becomes 3, -6, 30, 21, 12, 3, ...

6 means the sequence s(k) becomes 6, -30, -21, -12, -3, 6, ...

9 means the sequence s(k) oscillates between two numbers, each of which have a digit sum of 9. For example, 18 -> -63 -> 18 -> -63 -> ... so a(18) = 9.

11 means the sequence s(k) oscillates between two numbers, each of which have a digit sum of 11. For example, 65 -> -56 -> 65 -> ... so a(65) = 11.

a(n) = 2 for n = 2, 23, 62, 77, 119, 194, 287, 398. The next number n such that a(n) = 2 is over 10^5. This is believed to be finite.

a(n) = 11 for n = 8, 17, 26, 29, 35, 38, 47, 56, 65, 74, 83, 92, 149, 158, 167, 197. The next number n is over 10^5. This is believed to be finite.

The subsequences when a(n)=0, 3, 6, and 9 are believed to be infinite.

Examples

			a(19) = 0 because 19 - (1+9)^2 = -81. Then -81 + (8+1)^2 = 0.
a(13) = 6 because 13 - (1+3)^2 = -3. Then -3 + (3)^2 = 6.
a(17) = 11 because 17 - (1+7)^2 = -47. Then -47 + (4+7)^2 = 74. Then 74 - (7+4)^2 = -47, and so on.
a(23) = 2 because 23 - (2+3)^2 = -2. Then -2 + (2)^2 = 2.
a(25) = 3 because 25 - (2+5)^2 = -24. Then -24 + (2+4)^2 = 12. Then 12 - (1+2)^2 = 3.
a(28) = 9 because 28 - (2+8)^2 = -72. Then -72 + (7+2)^2 = 9. Then 9-(9)^2 = -72, and so on.
		

Crossrefs

Cf. A007953, A003132.

Programs

PARI

a(n)=k=n;c=1;v=List();listput(v,k);while(c,if(k>=0,k-=sumdigits(k)^2;c+=1;if(k==2||k==3||k==0||k==6||k==9,return(k));if(vecsearch(Vec(v),k),return(sumdigits(abs(k))));listput(v,k));if(k<0,k+=sumdigits(-k)^2;c+=1;if(k==2||k==3||k==0||k==6||k==9,return(k));if(vecsearch(Vec(v),k),return(sumdigits(abs(k))));listput(v,k));c+=1)