Another Palindromic Cyclops Number

A thousand and ten days ago, I made a post titled Palindromic Cyclops Numbers in which I focused on the number associated with my then diurnal age of 26062. Today I turned 27072 days old and I'm reminded of such numbers once again. They form OEIS  A138131:

A138131

Palindromic cyclops numbers.

The sequence follow a predictable pattern:

0, 101, 202, 303, 404, 505, 606, 707, 808, 909, 11011, 12021, 13031, 14041, 15051, 16061, 17071, 18081, 19091, 21012, 22022, 23032, 24042, 25052, 26062, 27072, 28082, 29092, 31013, 32023, 33033, 34043, 35053, 36063, 37073, 38083

Some of these numbers are prime, such as 101, but not all. 27072 is composite and in fact it has 42 divisors which qualifies it for membership of OEIS A175750$$27072 = 2^6 \times 3^2 \times 47 \rightarrow \text{ 42 divisors}$$

A175750

Numbers with 42 divisors.

The initial members of the sequence are as follows:

2880, 4032, 4800, 6336, 7488, 9408, 9792, 10944, 11200, 13248, 14580, 15552, 15680, 16704, 17600, 17856, 20412, 20800, 21312, 23232, 23328, 23616, 24768, 27072, 27200, 30400, 30528, 32076, 32448, 33984, 34496, 35136, 36450, 36800, 37908, 38592, 38720, 40768

I've marked in red the numbers in this sequence that are palindromic. As can be seen, 27072 is the only palindromic cyclops number that has 42 divisors. If we extend the range up to ten million, there are only these very few palindromic numbers with 42 divisors: 2308032, 4099904, 6714176 and 8820288.

Another of 27072's claim to fame is that it is 100-gonal number and thus a member of OEIS A261276. Such numbers are generated from the formula$$ \begin{align} \text{number } &= \frac{(s-2) \times n \times(n-1)}{2}+ n \\ &= \frac{98 \times n \times(n-1)}{2}+ n \text{   since } s=100 \end{align} $$The initial members of the sequence are (permalink):

0, 1, 100, 297, 592, 985, 1476, 2065, 2752, 3537, 4420, 5401, 6480, 7657, 8932, 10305, 11776, 13345, 15012, 16777, 18640, 20601, 22660, 24817, 27072, 29425, 31876, 34425, 37072, 39817, 42660, 45601, 48640, 51777, 55012, 58345, 61776, 65305, 68932, 72657, 76480

The number 27072 is also "bipronic" which is a term I've not encountered before, although I'm familiar with the term "pronic". The former term is an extension of the later so that bipronic numbers are of the form:$$ x \times (x+1) \times y \times (y+1) \\ \text{ where }x \text{ and } y \text{ are distinct integers}$$In the case of 27072 we have$$27072 = 3 \times 4 \times 47 \times 48$$These types of bipronic numbers form OEIS A053990 and if we relax the condition that \(x\) and \(y\) need to be distinct, then we have OEIS A072389. Combining the bipronics from OEIS A072389 with the palindromes, we get OEIS A346919:

A346919

Numbers that are both palindromes (A002113) and terms of A072389.

The initial members of this sequence are:

0, 4, 252, 2112, 2772, 6336, 21012, 27072, 42924, 48384, 48984, 63036, 252252, 297792, 407704, 2327232, 2572752, 2747472, 2774772, 2958592, 4457544, 4811184, 6378736, 6396936, 25777752, 27633672, 29344392, 63099036, 63399336, 404080404, 409757904, 441525144

The number 27072 also arises when we consider the sum of the divisors of the number of partitions of \(n\). These sums form OEIS A139041:

A139041

Sum of divisors of the number of partitions of \(n\).

The initial members of this sequence are:

1, 3, 4, 6, 8, 12, 24, 36, 72, 96, 120, 96, 102, 240, 372, 384, 480, 576, 1026, 960, 2340, 2016, 1512, 3224, 3240, 6720, 6336, 6588, 6048, 13104, 11232, 12768, 17784, 22176, 22344, 17978, 27072, 35112, 69696, 87552, 74496, 87048, 104544, 97216, 137088, 214896

In the case of 27072, we have \(n=37\) with the the number of partitions of 37 being equal to 21637 and with the divisors [1, 7, 11, 77, 281, 1967, 3091, 21637] adding to the number. There are of course many other properties associated with this number but I'll end off there.