Personal Investigation: Part Three

This post builds on my two previous posts so it's best to look at those first in order to properly understand what I'm doing. These posts are:

• Personal Investigation: Part 1
• Personal Investigation: Part 2

Looking at the maximal values associated with progressively summing the squares of dyads, triads, tetrads etc. and then finding the sum of their digits, the need for a consistent notational system arose. I wanted to be able to describe that specific term in the sequence with the maximum value for a particular $k$. Thus I came up with $a_{k_{max, \, N}}$ for this maximum value where $k$ is the exponent and $N$ is the number of terms that are being added together. Let's look at some specific examples:

DYAD

$a_{k_{max, \, N=2}}$ can be used to represent the maximum value attained by $a_n$ for a particular $k$ when summing two terms progressively, where: $$a_n= \text{ sum of digits } \big (a_{n-1}^k+a_{n-2}^k \big ) \text { with }a_0=0, a_1=1$$ The sequence for $2 \leq k \leq 50$ is as follows:

[14, 28, 35, 45, 56, 91, 83, 110, 98, 135, 128, 151, 155, 181, 197, 218, 200, 241, 275, 271, 296, 286, 308, 319, 341, 351, 350, 376, 353, 410, 443, 433, 395, 495, 443, 501, 551, 521, 548, 565, 614, 620, 614, 604, 611, 646, 641, 716, 701]

So we can write $a_{2_{max, \, N=2}}=14$,  $a_{3_{max, \, N=2}}=28$ etc.

TRIAD

$a_{k_{max, \, N=3}}$ can be used to represent the maximum value attained by $a_n$ for a particular $k$ when summing three terms progressively, where: $$a_n= \text{ sum of digits } \big (a_{n-1}^k+a_{n-2}^k +a_{n-3}^k\big ) \text { with }a_0=0, a_1=1,a_2=2$$ The sequence for $2 \leq k \leq 50$ is:

[24, 34, 35, 58, 66, 93, 98, 99, 117, 140, 147, 165, 191, 181, 197, 218, 236, 248, 266, 297, 297, 311, 309, 347, 344, 362, 360, 408, 416, 436, 425, 449, 479, 482, 479, 544, 546, 546, 566, 570, 597, 624, 632, 702, 665, 664, 696, 710, 725]

 So we can write $a_{2_{max, \, N=3}}=24$,  $a_{3_{max, \, N=3}}=34$ etc.

TETRAD

$a_{k_{max, \, N=4}}$ can be used to represent the maximum value attained by $a_n$ for a particular $k$ when summing four terms progressively, where: $$a_n= \text{ sum of digits } \big (a_{n-1}^k+a_{n-2}^k +a_{n-3}^k+a_{n-4}^k\big )$$ $$\text { with }a_0=0, a_1=1, a_2=2, a_3=3$$ The sequence for $2 \leq k \leq 50$ is:

[26, 28, 47, 56, 57, 82, 99, 118, 119, 147, 173, 183, 188, 206, 209, 214, 236, 282, 263, 271, 297, 311, 335, 338, 371, 357, 389, 409, 399, 442, 459, 476, 485, 457, 525, 541, 575, 567, 566, 633, 579, 651, 659, 666, 666, 699, 722, 762, 764]

So we can write $a_{2_{max, \, N=4}}=26$,  $a_{3_{max, \, N=4}}=28$ etc. 

PENTAD

$a_{k_{max, \, N=5}}$ can be used to represent the maximum value attained by $a_n$ for a particular $k$ when summing five terms progressively, where: $$a_n= \text{ sum of digits } \big (a_{n-1}^k+a_{n-2}^k +a_{n-3}^k+a_{n-4}^k+a_{n-5}^k\big )$$ $$\text { with }a_0=0, a_1=1, a_2=2, a_3=3, a_4=4$$ The sequence for $2 \leq k \leq 50$ is:

[26, 37, 38, 63, 65, 98, 92, 118, 122, 147, 156, 186, 197, 199, 198, 234, 255, 266, 281, 306, 324, 324, 335, 362, 357, 380, 405, 408, 426, 458, 474, 496, 504, 510, 516, 559, 557, 577, 563, 604, 588, 628, 657, 667, 695, 709, 686, 754, 764]

So we can write $a_{2_{max, \, N=5}}=26$,  $a_{3_{max, \, N=5}}=37$ etc.  

HEXAD

$a_{k_{max, \, N=6}}$ can be used to represent the maximum value attained by $a_n$ for a particular $k$ when summing six terms progressively, where: $$a_n= \text{ sum of digits } \big (a_{n-1}^k+a_{n-2}^k +a_{n-3}^k+a_{n-4}^k+a_{n-5}^k+a_{n-6}^k\big )$$ $$\text { with }a_0=0, a_1=1, a_2=2, a_3=3, a_4=4,a_5=4$$ The sequence for $2 \leq k \leq 50$ is:

[26, 39, 47, 66, 78, 91, 98, 118, 131, 147, 167, 175, 191, 207, 234, 232, 230, 269, 281, 288, 311, 335, 329, 355, 380, 393, 452, 412, 437, 440, 464, 486, 509, 510, 528, 544, 585, 575, 576, 608, 617, 630, 654, 656, 684, 705, 725, 734, 764]

So we can write $a_{2_{max, \, N=6}}=26$,  $a_{3_{max, \, N=6}}=39$ etc.  

To be continued: