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A109924 Least palindromic multiple of concatenation 123...n.

%I A109924 #27 Aug 26 2024 11:43:20
%S A109924 1,252,8118,28382,536797635,6180330816,85770307758,2889123219882,
%T A109924 535841353148535,135444949494445310,1522312136776312132251,
%U A109924 2111913320628668260233191112,6690072525779588859775252700966,202511080654222947749222456080115202,538412926804799527505725997408629214835
%N A109924 Least palindromic multiple of concatenation 123...n.
%C A109924 When n is a multiple of 10, any multiple of 123...n has trailing zeros, therefore it cannot be palindromic. The terms listed as a(10k) are therefore the least palindromic multiples with "invisible leading zeros allowed", or equivalently, trailing zeros ignored.
%C A109924 Subsequence of A020485.
%H A109924 P. De Geest, <a href="https://www.worldofnumbers.com/em36.htm">Smallest multipliers to make a number palindromic</a>.
%e A109924 123*j is not palindromic for j < 66 and 123*66 = 8118, hence a(3) = 8118.
%t A109924 f[n_] := Block[{k = 1, p = FromDigits[ Flatten[ IntegerDigits /@ Range[n]]]}, While[ If[ Mod[p, 10] == 0, p/=10]; While[k*p != FromDigits[ Reverse[ IntegerDigits[k*p]]], k++ ]]; k*p]; Table[ f[n], {n, 11}] (* _Robert G. Wilson v_, Jul 19 2005 *)
%o A109924 (PARI) intreverse(n) = local(d, rev); rev=0; while(n>0, d=divrem(n, 10); n=d[1]; rev=10*rev+d[2]);
%o A109924 {s="";for(n=1,10,s=concat(s,n);k=eval(s);if(n%10==0,m=0, j=1;while((m=k*j)!=intreverse(m),j++));print1(m,","))}
%o A109924 (PARI) A109924(n)={ n=eval(concat(vector(n,i,Str(i))));forstep(i=n/10^valuation(n,10),9e99,n/10^valuation(n,10), (m=Vec(Str(i)))==vecextract(m,"-1..1")&return(i*10^valuation(n,10)))} \\ _M. F. Hasler_, Jun 19 2011
%Y A109924 Cf. A020485, A050782, A061816, A109929.
%K A109924 nonn,base
%O A109924 1,2
%A A109924 _Amarnath Murthy_, Jul 16 2005
%E A109924 Edited and extended (a(5) to a(10)) by _Klaus Brockhaus_, Jul 19 2005
%E A109924 a(10)-a(11) from _Robert G. Wilson v_, Jul 19 2005
%E A109924 Definition of a(10k) clarified by _M. F. Hasler_, Jun 19 2011.
%E A109924 a(12)-a(14) from _Giovanni Resta_, Sep 22 2019
%E A109924 a(15) from _Giovanni Resta_, Sep 24 2019