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A069560 Squares in which the k-th significant digit either divides k or is a multiple of k.

%I A069560 #16 Sep 20 2024 12:27:10
%S A069560 1,4,9,16,25,49,64,81,121,144,169,324,361,625,961,1369,1681,2116,2916,
%T A069560 4624,8649,11664,12321,14161,14641,51984,114921,151321,214369,311364,
%U A069560 351649,1151329,1252161,1658944,7311616,7354944,41254929,41654116
%N A069560 Squares in which the k-th significant digit either divides k or is a multiple of k.
%C A069560 If the smallest prime divisor of n is > 7 then the n-th digit is 1.
%C A069560 a(65) if it exists is > 10^50. - _Andrew Howroyd_, Sep 20 2024
%H A069560 Andrew Howroyd, <a href="/A069560/b069560.txt">Table of n, a(n) for n = 1..64</a>
%e A069560 8649 is a member in which the fourth digit is 8 a multiple of 4, the third one is 6 a multiple of 3, the second one is 4 a multiple of 2 and the least significant digit is 9.
%o A069560 (PARI) isok(k)={my(d=digits(k)); for(i=1, #d, my(t=d[#d+1-i]); if(!t || (t%i && i%t), return(0))); 1}
%o A069560 for(k=1, 10000, my(x=k^2); if(isok(x), print1(x, ", "))) \\ _Andrew Howroyd_, Sep 19 2024
%o A069560 (PARI) \\ faster program
%o A069560 B(k)={
%o A069560   local(L=List());
%o A069560   my(v=vector(k, i, select(t->t%i==0||i%t==0, [1..9])));
%o A069560   my(chk(d)=for(i=1, #d, if(!vecsearch(v[i], d[#d+1-i]), return(0)));1);
%o A069560   my(s=k\2, b=10^s);
%o A069560   my(recurse(i,m)=if(i==s,
%o A069560      for(j=sqrtint(m*b+b\10-1)+1, sqrtint(m*b+b-1), my(t=j^2); if(chk(digits(t%b)), listput(L,t))),
%o A069560     m*=10; foreach(v[i], t, self()(i-1, m+t))
%o A069560   ));
%o A069560   recurse(k, 0);
%o A069560   Vec(L);
%o A069560 }
%o A069560 concat(vector(12,k,B(k))) \\ _Andrew Howroyd_, Sep 19 2024
%Y A069560 Intersection of A000290 and A069570.
%Y A069560 Cf. A069556, A069557, A069558, A069559.
%K A069560 base,nonn
%O A069560 1,2
%A A069560 _Amarnath Murthy_, Mar 22 2002
%E A069560 More terms from _Sascha Kurz_, Mar 23 2002
%E A069560 Offset changed by _Andrew Howroyd_, Sep 19 2024