A077187 Smallest k such that the concatenation 123...(k-1) k (k-1)...321 ( a concatenation of natural numbers from 1 to k and back to 1) is a multiple of prime(n), or 0 if no such number exists.
0, 3, 0, 6, 2, 6, 109, 103, 100001, 1006, 17, 3, 5, 103, 1000002, 100012, 1019, 1002, 1001, 16, 8, 105, 1036, 104, 1002, 4, 100000000009, 100004, 52, 156, 10000000012, 1062, 8, 1002, 28, 102, 1011, 1000062, 30, 10001, 118, 52, 43, 10058, 34, 47
Offset: 1
Examples
a(5) = 2 as 121 is a multiple of 11.
Programs
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PARI
{ a(n) = local(p,c10,z,u,v,l,k,lk,L,q); p=prime(n); c10=Mod(10,p); z=znorder(c10); u=v=Mod(1,p); l=1; k=2; L=List(); while(1, lk=1+log(k+0.1)\log(10); if(k==10^(lk-1), L=List()); if( u*c10^(l+lk)+k*c10^l+v==0, return(k)); q=0; t=[u,v,k%p,l%z]; for(j=1,#L,if(t==L[j], q=1+#L-j)); if(q, k+=((10^lk-1-k)\q)*q; L=List(), listput(L,t)); u=u*c10^lk+k; v+=k*c10^l; l+=lk; k++) } \\ Max Alekseyev, Sep 11 2009
Extensions
3 more terms from Erich Friedman, Aug 08 2005
Extended by Max Alekseyev, Sep 11 2009