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A344633 Lengths k of k-digit integers of the form 1, 12, 123, 1234, ... (A057137) which are divisible by k.

%I A344633 #30 Sep 09 2024 09:34:41
%S A344633 1,2,3,5,6,9,10,12,14,15,16,18,30,90,96,110,197,210,270,330,390,410,
%T A344633 630,810,930,959,990,1110,1170,1210,1230,1470,1710,1890,1956,2310,
%U A344633 2430,2530,2538,2710,2730,2790,2802,2922,2970,3330,3510,3519,3630,3690,4115,4245
%N A344633 Lengths k of k-digit integers of the form 1, 12, 123, 1234, ... (A057137) which are divisible by k.
%C A344633 It is easy to prove that 10*3^k, k >= 0 is always a solution.
%F A344633 {n: n|A057137(n)}. - _R. J. Mathar_, Aug 16 2021
%e A344633 3 is a term since 123 is divisible by 3 (123 = 3*41).
%p A344633 for n from 1 to 5000 do
%p A344633     if modp(A057137(n),n) = 0 then
%p A344633         printf("%d,",n) ;
%p A344633     end if;
%p A344633 end do: # _R. J. Mathar_, Aug 16 2021
%o A344633 (Python)
%o A344633 a ="1234567890"
%o A344633 for k in range(10):
%o A344633     a = a + a
%o A344633 sol = ""
%o A344633 for n in range(1, len(a)):
%o A344633     if int(a[0:n]) % n == 0:
%o A344633         sol = sol + str(n) + ", "
%o A344633 print(sol)
%o A344633 (PARI) f(n) = 137174210*10^n\1111111111; \\ A057137
%o A344633 isok(k) = (f(k) % k) == 0; \\ _Michel Marcus_, Aug 16 2021
%Y A344633 Cf. A057137.
%K A344633 nonn,base
%O A344633 1,2
%A A344633 _Reiner Moewald_, May 26 2021