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Thursday, 10 February 2022

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 3×266113 contains 333 as a substring:3×266113=565333_65411393

I've been mistakenly thinking that the number was the exponent and that I was dealing with 3×326611. Naturally, with such an enormous number, it would be likely that 333 would occur. Now that I've recognised my error, I'm creating this post to make amends for my neglect. In general: a super-d number is a number n for d=2,,9 such that dnd contains a substring made of d digits of d
The first super-2 number is 19 where 2×192=722_ and the first super-3 number is 261 where 3×2613=5333_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:3×471003=313461333_000000
Figure 3 shows that the spiral pattern of super-d numbers up to 2502 contains some long runs of consecutive terms.

Figure 3: source

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