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A386501 Numbers k divisible by A004719(k), excluding trivial cases.

%I A386501 #23 Aug 03 2025 16:09:01
%S A386501 105,108,405,1001,1005,1008,1020,2002,2025,2040,2050,3003,3060,4004,
%T A386501 4005,4080,5005,6006,6075,7007,7050,8008,9009,10005,10008,10020,10032,
%U A386501 10065,10098,10101,10125,10206,10250,16005,19008,20007,20025,20040,20050
%N A386501 Numbers k divisible by A004719(k), excluding trivial cases.
%C A386501 Trivial cases are identified as (1) values of k where there are already no 0s besides leading 0s, like 255 or 1296, such that A004719(k)=k, or (2) where k mod 10 = 0 and k/10 is already in the sequence or is itself a trivial case, like 10080 or 2550. In case (1), k/A004719(k) is equivalent to k/k (as in 255/255). In case (2), k/A004719(k) = 10 * (k/10)/A004719(k/10) when we already know that (k/10)/A004719(k/10) is already an integer (as in 1080/18).
%C A386501 Any number k of the form 1|(at least one 0)|5, such as 105 or 10000000005, will be included in this sequence because k will always be divisible by 3 and 5 due to divisibility rules, and thus will be divisible by A004719(k)=15.
%C A386501 Numbers of form 1|(at least one 0)|8, such as 108 or 10000008, or 4|(at least one 0)|5, such as 405 or 400005, will be included in this sequence for similar reasons.
%e A386501 A004719(108)=18, 108/18=6.
%e A386501 A004719(9009)=99, 9009/99=91.
%e A386501 A004719(2040)=24, 2040/24=85, 2040 is nontrivial because 204/24=17/2.
%e A386501 50 is trivial because 50/10 = 5, and 5 is trivial because A004719(5)=5.
%o A386501 (Python)
%o A386501 def removeZeros(number):
%o A386501     stringNum = str(number)
%o A386501     stringNum = stringNum.replace("0", "")
%o A386501     return int(stringNum)
%o A386501 for x in range(1, 100000):
%o A386501     smallInt = removeZeros(x)
%o A386501     if smallInt == x:
%o A386501         continue
%o A386501     if x % smallInt == 0:
%o A386501         if x % 10 == 0:
%o A386501             if (x//10) % removeZeros(x//10) == 0:
%o A386501                 continue
%o A386501         print(x)
%Y A386501 Subset of A090055.
%Y A386501 Cf. A004719, A052382.
%Y A386501 Cf. A096093, A090053.
%K A386501 nonn,base
%O A386501 1,1
%A A386501 Anuraag Pasula and _Walter Robinson_, Jul 23 2025