A292684 a(n) is the number of positive integers k not divisible by 10 such that f(kN) = f(N) for N = A292683(n) and f(x) = x / (x without its first digit: A217657(x)).

9, 4, 1, 9, 9, 4, 3, 3, 3, 3, 1, 1, 9, 9, 9, 7, 4, 9, 9, 9, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 9, 9, 9, 9, 9, 1, 1, 1, 1, 1, 1, 9, 9, 9, 9, 3, 9, 9, 9, 9, 9, 7, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 9, 9, 9, 9, 9, 9, 9

Offset: 1

Author

M. F. Hasler, Oct 17 2017

Keywords

• nonn
• base

Comments

Sequence A292683 lists the numbers n which are divisible by A217657(n), i.e., by n with its first digit removed.

We exclude k with trailing 0's (just like in A292683) because if k*N has the property, then 10*k*N trivially also has the property.

Is there any number for which there are more than 9 possible k-values?

All of the k-values are listed in the table A292685.

Examples

			For A292683(1) = 11, we have k = 1, ..., 9 satisfying 11*k / A217657(11*k) = 11.
For A292683(2) = 12, we have k = 1, 2, 3, 4 satisfying 12*k / A217657(12*k) = 6.
For A292683(3) = 15, we have only k = 1 satisfying 15*k / A217657(15*k) = 3.
For A292683(4) = 21, we have k = 1, 2, 3, 4, 5, 15, 25, 35 and 45 satisfying 21*k / A217657(21*k) = 2.
		

Crossrefs

Cf. A292683, A292685, A217657, A000030.

Programs

PARI

(A217657(n)=n%10^logint(n,10)); A292684(n,N=A292683(n),r=N/A217657(N),a=[1])={for(k=2,oo,k%10||next;k>10*a[#a]&&break;A217657(k*N)*r==k*N&&a=concat(a,k));#a} \\ Instead of the 1st arg. n, one can directly give N (= A292683(n) by default) as 2nd arg. One could store only the last 'a' (and increase a counter) instead of storing all 'a's.