Higher Order Smith Numbers

𝕾𝖒𝖎𝖙𝖍 𝕹𝖚𝖒𝖇𝖊𝖗𝖘

I delved into Smith numbers in a post titled Smith Numbers and Repunits on April 21st 2016. In that post I explained that:

A Smith number is defined by Wikipedia as a composite number for which, in a given base (in base 10 by default), the sum of its digits is equal to the sum of the digits in its prime factorization. For example, 378 = 2 × 3 × 3 × 3 × 7 is a Smith number since 3 + 7 + 8 = 2 + 3 + 3 + 3 + 7. In this definition the factors are treated as digits: for example, 22 factors to 2 × 11 and yields three digits: 2, 1, 1. Therefore 22 is a Smith number because 2 + 2 = 2 + 1 + 1.

The first few Smith numbers are:

4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438, 454, 483, 517,526, 535, 562, 576, 588, 627, 634, 636, 645, 648, 654, 663, 666, 690, 706, 728, 729, 762, 778, 825, 852, 861, 895, 913, 915, 922, 958, 985, 1086 … (sequence A006753 in OEIS)

Smith numbers were named by Albert Wilansky of Lehigh University. He noticed the property in the phone number (493-7775) of his brother-in-law Harold Smith:$$ \begin{align} 4937775 &= 3 \times 5 \times 5 \times 65837 \\ &\rightarrow 4 + 9 + 3 + 7 + 7 + 7 + 5 \\&= 3 + 5 + 5 + 6 + 5 + 8 + 3 + 7 \\ &= 42 \end{align} $$𝕾𝖒𝖎𝖙𝖍 𝕹𝖚𝖒𝖇𝖊𝖗𝖘 𝖔𝖋 𝕺𝖗𝖉𝖊𝖗 𝕿𝖍𝖗𝖊𝖊

Today I turned 26637 days old and, as it turns out, this is a Smith number since 26637 = 3 x 13 x 683 and the sum of its digits (24) coincides with the sum of the digits of its prime factors. However, as I discovered, it is also a Smith number of order 3 and a member of OEIS A178213. The OEIS comments explain that:

Smith numbers of order 3 are composite numbers \(n\) not in A176670 such that the sum of the cubes of the digits of \(n\) equals the sum of the cubes  of the digits of the prime factors of \(n\) (with multiplicity). 

Now the members of OEIS A176670 (referred to above) are composite numbers having the same digits as their prime factors (with multiplicity), excluding zero digits. An example is 25105 = 5 x 5021 in which the number and the factorization of the number have digits 1, 2, 5, 5 when sorted and excluding zeroes. It is these sorts of numbers that are excluded from OEIS A178213, even though they are Smith numbers. 

Now getting back to OEIS A178213 and Smith numbers of order 3, the initial members are:

6606, 8540, 13086, 16866, 21080, 26637, 27468, 33387, 34790, 35364, 35377, 40908, 44652, 48154, 48860, 52798, 54814, 55055, 57726, 57894, 66438, 67297, 67356, 67594, 69549, 72465, 72598, 73026, 74371, 74785, 77485, 78745, 81546, 83175, 85927, 90174, 91208, ...

Let's use \(26637 = 3 \times 13 \times  683\) as an example:$$ \begin{align} 26637 &\rightarrow 2^3+6^3+6^3+3^3+7^3 \\&= 3^3+1^3+3^3+6^3+8^3+3^3 \\ &= 810 \end{align} $$Here is a permalink to a Sage Math algorithm that will generate the members of this sequence.

𝕾𝖒𝖎𝖙𝖍 𝕹𝖚𝖒𝖇𝖊𝖗𝖘 𝖔𝖋 𝕺𝖗𝖉𝖊𝖗 𝕿𝖜𝖔

Of course if there can be Smith numbers of order 3, there can be Smith numbers of order 2 and these constitute OEIS A174460 whose initial members are:

56, 58, 810, 822, 1075, 1519, 1752, 2145, 2227, 2260, 2483, 2618, 2620, 3078, 3576, 3653, 3962, 4336, 4823, 4974, 5216, 5242, 5386, 5636, 5719, 5762, 5935, 5998, 6220, 6424, 6622, 6845, 7015, 7251, 7339, 7705, 7756, 8460, 9254, 9303, 9355, 10481, 10626, 10659, ...

The example is given of:$$ \begin{align} \text{a}(2) &= 58 \\ &= 2 \times 29 \\ 58 &\rightarrow 5^2 + 8^2 \\&= 2^2 + 2^2 + 9^2 \\ &= 89 \end{align} $$𝕾𝖒𝖎𝖙𝖍 𝕹𝖚𝖒𝖇𝖊𝖗𝖘 𝖔𝖋 𝕺𝖗𝖉𝖊𝖗 𝕱𝖔𝖚𝖗

Clearly these 2nd order numbers are more frequent than their 3rd order counterparts. What about Smith numbers of order 4? These constitute OEIS A178193 and are not numerous. The initial members are:

3777, 7773, 17418, 30777, 53921, 66111, 97731, 111916, 119217, 122519, 128131, 133195, 135488, 138878, 145229, 178814, 180174, 198581, 257376, 269636, 281179, 296396, 317686, 358256, 362996, 366514, 394114, 435777, 457377, 469552, 475856, 502960, 513833

The example is given of:

\(3777 = 3 \times 1259 \) is composite; sum of 4th power of the digits is \(3^4 + 7^4 + 7^4 + 7^4 = 7284\). Sum of 4th power of the digits of the prime factors 3, 1259 is \(3^4 + 1^4 + 2^4 + 5^4 + 9^4 = 7284\). The sums are equal, so 3777 is in the sequence.

𝕾𝖒𝖎𝖙𝖍 𝕹𝖚𝖒𝖇𝖊𝖗𝖘 𝖔𝖋 𝕺𝖗𝖉𝖊𝖗 𝕱𝖎𝖛𝖊

The Smith numbers of order 5 are even less numerous and constitute OEIS A178203. The initial members are:

414966, 443166, 454266, 1274664, 1371372, 1701856, 1713732, 1734616, 1771248, 1858436, 1858616, 2075664, 2624976, 3606691, 3771031, 3771301, 4266914, 4414866, 4461786, 4605146, 4670576, 4710739, 5209663, 5281767, 5434572, 5836565, 5861712, 5871968, 6046357 

The example is given of: $$ \begin{align} \text{a}(10) &= 1858436 \\ &= 2 \times 2 \times 29 \times 37 \times 433 \\ &\rightarrow 1^5 + 3^5 + 4^5 + 5^5 + 6^5 + 2 \times 8^5 \\&= 3 \times 2^5 + 3 \times 3^5 + 4^5 + 7^5 + 9^5 \\ &= 77705 \end{align} $$𝕾𝖒𝖎𝖙𝖍 𝕹𝖚𝖒𝖇𝖊𝖗𝖘 𝖔𝖋 𝕺𝖗𝖉𝖊𝖗 𝕾𝖎𝖝

The OEIS also lists Smith numbers of order 6. These numbers constitute OEIS A178204. These are quite rare and the initial members are:

0844882, 113986781, 130852098, 141176320, 168137185, 170774472, 178180163, 181681157, 181693781, 183161897, 187117638, 215149451, 261666000, 284804842, 294557945, 307711074, 335524949, 337194240, 344552927, 347391040, 355318188, 358831104, 368657536

The example is given of:$$ \begin{align} \text{a}(4)& = 141176320 \\ &= 2^9 \times 5 \times 55147 \\ 141176320 &\rightarrow 1^6+4^6+1^6+1^6+7^6+6^6+3^6+2^6\\&= 9 \times 2^6+5^6+5^6+5^6+1^6+4^6+7^6 \\ &= 169197 \end{align} $$The permalink I've provided can be easily modified to accommodate Smith numbers of any order but may well time out for higher orders.