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A334916 a(n) is the smallest number > 1 whose base n digits yield the original number when added and multiplied left to right; or 0 if no such number exists.

%I A334916 #17 Jul 23 2025 16:01:53
%S A334916 0,0,0,6,12,160,324,405,12,8385,36,189,784,32,1656,20,721,25215,80,45,
%T A334916 559,2585,5525,323844,30,160,60,90,150,1071,11650,1038448,6275,2669,
%U A334916 77,42,2224,324224,1817,2016,252,7425,1593074855,96,5450,192,345906,23541,56
%N A334916 a(n) is the smallest number > 1 whose base n digits yield the original number when added and multiplied left to right; or 0 if no such number exists.
%C A334916 These numbers have been called "baseless in base n".
%C A334916 a(n) is divisible by its last base n digit.
%C A334916 The number 8385 = ((((8)8+3)3+8)8+5)5 is known to be the unique baseless number in base 10. Are there number bases n, other than 6 and 10, that have a unique example?
%C A334916 If the term a(107) is not zero, then it is at least a(107) > 107^6 > 1.5*10^12. Is it true that a(n)>0 for all n>3?
%H A334916 Math StackExchange user "Vepir" (Matej Veselovac), <a href="https://pastebin.com/raw/ydA64hig">Terms a(n) < 10^10 for n < 500, and including the 11th record a(73) ~ 2*10^10.</a>
%H A334916 Math StackExchange, <a href="https://math.stackexchange.com/q/3658214/318073">Does every number base have at least one "baseless number"?</a>
%F A334916 If n is a perfect square, then a(n) = n + sqrt(n). Otherwise, a(n) > 2n.
%e A334916 Every number can be written as A = (...((((a)N+b)N+c)N+d)...) where a,b,c,d,... are digits of number A in base N. If we take that expression and replace the "multiplications by base N" with "multiplications by digits a,b,c,d,..." and also multiply it with the last digit to use up all digits, we get some number A*. If it holds A = A*, then we say number A is a baseless number.
%e A334916 For example, the decimal number base has only one baseless number:
%e A334916 .
%e A334916 a(10) = 8385 = ((((8)*10+3)*10+8)*10+5) = ((((8)*8+3)*3+8)*8+5)*5.
%e A334916 .
%e A334916 There are at most finitely many baseless numbers for every fixed number base. For example, the number base 4 has exactly three baseless numbers:
%e A334916 .
%e A334916 6 = ((1)*4+2)        = ((1)*1+2)*2       = 12_4;
%e A334916 27 = (((1)*4+2)*4+3) = (((1)*1+2)*2+3)*3 = 123_4;
%e A334916 46 = (((2)*4+3)*4+2) = (((2)*2+3)*3+2)*2 = 232_4;
%e A334916 .
%e A334916 The smallest of them is 6, hence a(4)=6.
%o A334916 (C++)
%o A334916 #include <iostream>
%o A334916 using namespace std;
%o A334916 typedef unsigned long long ull;
%o A334916 int main() {
%o A334916     for (int b = 1; b>0; b++) {
%o A334916     for (ull n = b; n>0; n++) {
%o A334916             if (b<4) {cout << b << " " << 0 << endl;break;}
%o A334916             if (b==43) {cout << b << " " << 1593074855 << endl;break;}
%o A334916             if (b==73) {cout << b << " " << 25683204625 << endl;break;}
%o A334916             ull a=n, m=n;
%o A334916             while (m != 0) {
%o A334916                 int d = a%b;
%o A334916                 if (d>0 && m%d==0) {
%o A334916                     m /= d; if (m < d) {break;} m -= d; a -= d; a /= b;
%o A334916                 } else {break;}
%o A334916             }
%o A334916             if (m==0 && a==0){cout << b << " " << n << endl;break;}
%o A334916     }}
%o A334916     return 0;
%o A334916 }
%o A334916 (PARI) \\ for n>=4
%o A334916 isok(k,n) = {my(d=digits(k, n), s=0); for (i=1, #d, s = (s+d[i])*d[i];); s == k;}
%o A334916 a(n) = {my(k=2); while (!isok(k, n), k++); k;} \\ _Michel Marcus_, Jun 18 2020
%Y A334916 Cf. A000290 (perfect squares), A334917 (indices of records).
%K A334916 nonn,base,hard
%O A334916 1,4
%A A334916 _Matej Veselovac_, May 16 2020