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:$$ \text{ a super-d number is a number } n \text{ for }d=2, \dots ,9\\ \text{ such that  } d\cdot n^d \text{ contains a substring made of } d  \text{ digits of } d$$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