The number associated with my diurnal age today, 27384, has a special property that qualifies it for membership in a rather exclusive OEIS sequence.
A187584 | Least number divisible by at least \(n\) of its digits, different and > 1. |
Here are members of the sequence for the various values of \(n\):
- \(n =1 \rightarrow 2 =2\)
- \(n =2 \rightarrow 24 = 2^3 \times 3\)
- \(n =3 \rightarrow 248 = 2^3 \times 31\)
- \(n =4 \rightarrow 2364 = 2^2 \times 3 \times 197\)
- \(n =5 \rightarrow 27384 = 2^3 \times 3 \times 7 \times 163\)
- \(n =6 \rightarrow 243768 = 2^3 \times 3 \times 7 \times 1451\)
- \(n =7 \rightarrow 23469768=2^3 \times 3^2 \times 7 \times 46567\)
- \(n =8 \rightarrow 1234759680=2^{12} \times 3^3 \times 5 \times 7 \times 11 \times 29 \)
It can be seen that the final two members of the sequence, 23469768 and 1234759680, have eight and nine digits respectively whereas the earlier members have numbers of digits equal to \(n\). In the case of 27384, the five digits are 2, 3, 4, 7 and 8:
- \( \dfrac{27384}{2} =13692 \)
- \( \dfrac{27384}{3} = 9128\)
- \( \dfrac{27384}{4} = 6846\)
- \( \dfrac{27384}{7} = 3912\)
- \( \dfrac{27384}{8} = 3423\)
There are other OEIS sequences that list all the numbers divisible by at least \(n\) digits and these are:
- a(1)=2=A185186
- a(2)=24=A187516
- a(3)=248=A187398
- a(4)=2364=A187238
- a(5)=27384=A187533
- a(6)=243768=A187534
- a(7)=23469768=A187551
- a(8)=1234759680=A187565
27384, 29736, 36792, 37296, 37926, 38472, 46872, 73248, 73962, 78624, 79632, 84672, 92736, 123648, 123864, 123984, 124368, 126384, 129384, 132648, 132864, 132984, 134928, 136248, 136824, 138264, 138624, 139248, 139824, 142368, 143928, 146328, 146832, 148392, 148632, 149328, 149832, 162384, 163248, 163824, 164328, 164832, 167328, 167832, 168432, 172368, 183264, 183624, 184392, 184632, 186432, 189432, 192384, 193248, 193824, 194328
27384 is also a Lynch-Bell number. See my blog post titled Lynch-Bell Numbers.
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