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

105, 108, 405, 1001, 1005, 1008, 1020, 2002, 2025, 2040, 2050, 3003, 3060, 4004, 4005, 4080, 5005, 6006, 6075, 7007, 7050, 8008, 9009, 10005, 10008, 10020, 10032, 10065, 10098, 10101, 10125, 10206, 10250, 16005, 19008, 20007, 20025, 20040, 20050

Offset: 1

Author

Anuraag Pasula and Walter Robinson, Jul 23 2025

Keywords

• nonn
• base

Comments

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).

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.

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.

Examples

			A004719(108)=18, 108/18=6.
A004719(9009)=99, 9009/99=91.
A004719(2040)=24, 2040/24=85, 2040 is nontrivial because 204/24=17/2.
50 is trivial because 50/10 = 5, and 5 is trivial because A004719(5)=5.
		

Crossrefs

Subset of A090055.

Cf. A004719, A052382.

Cf. A096093, A090053.

Programs

Python

def removeZeros(number):
    stringNum = str(number)
    stringNum = stringNum.replace("0", "")
    return int(stringNum)
for x in range(1, 100000):
    smallInt = removeZeros(x)
    if smallInt == x:
        continue
    if x % smallInt == 0:
        if x % 10 == 0:
            if (x//10) % removeZeros(x//10) == 0:
                continue
        print(x)