A244188 Numbers n such that the digit that repeats the most at the end of n^k for some k is not the last digit of n.

2, 12, 22, 33, 37, 42, 52, 62, 72, 73, 77, 92, 102, 112, 113, 117, 122, 142, 152, 153, 162, 172, 192, 197, 202, 212, 222, 233, 237, 242, 252, 262, 272, 273, 277, 292, 302, 312, 313, 317, 322, 342, 352, 353, 362, 372, 392, 397, 402, 412, 422, 433, 437, 442, 452, 462, 472

Offset: 1

Author

Derek Orr, Jun 22 2014

Keywords

• nonn
• base

Comments

a(n) ends in either a 2, 3, or 7.

If a(n) ends in a 2, the digit that repeats itself the most at the end of a(n)^k is 8.

If a(n) ends in a 3, the digit that repeats itself the most at the end of a(n)^k is 7.

If a(n) ends in a 7, the digit that repeats itself the most at the end of a(n)^k is 3.

The numbers that end in 2 are congruent to {2, 12, 22, 42} mod 50.

The numbers that end in 3 are congruent to {33, 73, 113, 153} mod 200.

The numbers that end in 7 are congruent to {37, 77, 117, 197} mod 200.

Examples

			2^k ends in 2 of the same digit for k = 18 mod 20 (last digit is 4) and 19 mod 20 (last digit is 8). 2^k ends in 3 of the same digit for k = 39 mod 100 (last digit is 8). Since 8 is the only possibility, 8 must remain the only possibility for any larger run of identical digits. Since 8 is not the last digit of 2, then 2 is a member of this sequence.
33^k ends in 2 of the same digit for k = 1 mod 20 (last digit is 3) and 7 mod 20 (last digit is 7). 33^k ends in 3 of the same digit for k = 87 mod 100 (last digit is 7). Since 7 is the only possibility, 7 must remain the only possibility for any larger run of identical digits. Since 7 is not the last digit of 33, 33 is a member of this sequence.
		

Crossrefs

Cf. A243977, A244187, A243911, A243912.

Programs

PARI

seq(n)=for(m=2,6,cc=0;for(i=10^(m-1),10^m,st1=(n^i)%10^m;b="";for(j=1,m,b=concat(b,"1"));if(st1%eval(b)==0,for(d=i+1,10^m,sb1=(n^d)%10^m;if(sb1%eval(b)==0,if(sb1%10==st1%10,return(st1%10));if(sb1%10!=st1%10,cc++;break)))));if(cc==0,return(n%10)))
n=1;while(n<1000,if(seq(n)!=n%10,print1(n,", "));n++)