Super-6 Numbers

The number associated with my diurnal age yesterday (27257) is a super-6 number. This means that \(6 \times 27257^6\) has a run of six consecutive digits of 6 within it, specifically:$$6 \times 27257^6=2460478505381\underline{666666}506497894$$This number is the first member of the sequence OEIS A032746. The initial members are:

27257, 272570, 302693, 323576, 364509, 502785, 513675, 537771, 676657, 678146, 731378, 831122, 836553, 913797, 920456, 921269, 1045361, 1144983, 1169054, 1283069, 1288697, 1292673, 1343642, 1346117, 1472078, 1523993, 1640026

27257 is the first super-6 number

The generalization of super-6 numbers is super-\(d\) numbers and I've written about these types of numbers in posts titled Super-d Numbers Revisited on March 25th 2023 and also Super-d Numbers on February 10th 2022. Permalink.

Super-d Numbers Revisited

For some reasons, a search for super-\(d\) numbers failed to initially discover a previous post on the topic from February 22nd, 2022. Consequently, some of the content in that post has been repeated. Here is the earlier post titled Super-d Numbers. It's a good idea to view both posts as each contains certain content that isn't repeated in the other. The number of posts in this blog now exceeds 500 so it's easy to forget about previous posts. I need to be thorough in the tags that I add to each post.

Here is the new post created when I wasn't aware of the earlier post. 

For \(d=2, \dots,9\), a super-\(d\) number is a number \(n\) such that \(d \cdot n^d\) contains a substring made of \(d\) digits \(d\). For example, 261 is a super-3 number since \(3\cdot261^3=5\underline{333}8743\).

I was reminded of these numbers because my diurnal age today, 27019, is a super-2 number since \(2 \cdot 27019^2=14600527\underline{22}\). Figure 1 shows a table of the initial \(d\)-numbers for values of \(d\) from 2 to 9.

Figure 1:  source

Figure 2 shows the initial palindromic super-\(d\) numbers for values of \(d\) from 2 to 6.

Figure 2: source

The frequency of super-\(d\) numbers decreases as the value of \(d\) increases. The numbers in the range up to 40,000 are 4377, 420, 43, 12, 1, 0, 0, 0 for \(d\) = 2, 3, 4, 5, 6, 7, 8, 9 respectively. For me, a forthcoming super-6 number, and the only one is the range up to 40,000, is 27257 with the property that:$$6 \cdot 27257 \, ^6=2460478505381 \underline{666666} 506497894 $$Here is a Permalink to the calculation.

Super-d Numbers

So-called super-d numbers keep popping up in Numbers Aplenty from time to time in specific forms like super-2 numbers, super-3 numbers etc. I've ignored them for reasons that I'll explain later. Today I turned 26611 days old and one the properties of this number is that it's a super-3 number meaning that \( {\small 3 \times 26611^3} \) contains \( {\small 333} \) as a substring:$$3 \times 26611^3=565 \underline{333}65411393$$I've been mistakenly thinking that the number was the exponent and that I was dealing with \( {\small 3 \times 3^{26611}} \). Naturally, with such an enormous number, it would be likely that \( {\small 333} \) would occur. Now that I've recognised my error, I'm creating this post to make amends for my neglect. In general:$$ \begin{align} \text{ a super-d number is a number } n \text{ for }d=2, \dots ,9\\ \text{ such that  } d\cdot n^d \text{ contains a substring made of } d  \text{ digits of } d \end{align}$$The first super-2 number is 19 where \( {\small 2 \times 19^2=7\underline{22} }\) and the first super-3 number is 261 where \( {\small 3 \times 261^3=5\underline{333}8743 }\).

Figure 1 shows a list of the initial super-d numbers:

Figure 1: source

Up to 1000 the super-d numbers are:

19, 31, 69, 81, 105, 106, 107, 119, 127, 131, 169, 181, 190, 219, 231, 247, 261, 269, 281, 310, 318, 319, 331, 332, 333, 334, 335, 336, 337, 338, 339, 348, 369, 381, 419, 431, 454, 462, 469, 471, 481, 511, 519, 531, 558, 569, 581, 601, 619, 631, 669, 679, 681, 690, 715, 719, 731, 739, 749, 753, 769, 781, 782, 783, 784, 810, 819, 831, 869, 881, 919, 928, 931, 944, 969, 981, 988

Figure 2 shows the first few palindromic super-d number for small d:

Figure 2: source

It has been shown that all numbers ending in 471, 4710, or 47100 are super-3 numbers. For example:$$3 \times 47100^3=313461\underline{333}000000 $$Figure 3 shows that the spiral pattern of super-d numbers up to \( {\small 250^2} \) contains some long runs of consecutive terms.

Figure 3: source