An Unhappy Family

Today I turned 25634 days old and at first glance I found little of interest about the number after consulting my usual sources: the OEIS (Online Encyclopaedia of Integer Sequences) and Numbers Aplenty. However, after a little thought, I realised that the number is composed of the consecutive digits 2, 3, 4, 5 and 6. It thus belongs to a family of 120 numbers that are all composed of these five digits.

The members are:

23456, 23465, 23546, 23564, 23645, 23654, 24356, 24365, 24536, 24563, 24635, 24653, 25346, 25364, 25436, 25463, 25634, 25643, 26345, 26354, 26435, 26453, 26534, 26543, 32456, 32465, 32546, 32564, 32645, 32654, 34256, 34265, 34526, 34562, 34625, 34652, 35246, 35264, 35426, 35462, 35624, 35642, 36245, 36254, 36425, 36452, 36524, 36542, 42356, 42365, 42536, 42563, 42635, 42653, 43256, 43265, 43526, 43562, 43625, 43652, 45236, 45263, 45326, 45362, 45623, 45632, 46235, 46253, 46325, 46352, 46523, 46532, 52346, 52364, 52436, 52463, 52634, 52643, 53246, 53264, 53426, 53462, 53624, 53642, 54236, 54263, 54326, 54362, 54623, 54632, 56234, 56243, 56324, 56342, 56423, 56432, 62345, 62354, 62435, 62453, 62534, 62543, 63245, 63254, 63425, 63452, 63524, 63542, 64235, 64253, 64325, 64352, 64523, 64532, 65234, 65243, 65324, 65342, 65423, 65432

Looking at the graph above, it can be seen that there must be 120 possible paths joining all five vertices.

This provided an opportunity to investigate some of the properties of this family. Specifically, I explored how many members of the family were:

• prime
• semiprime
• sphenic

I also looked at how many members contained the factors 2 and 7, given that the prime factors of 25634 are 2, 7 and 1831.

To begin with only six members of the family are prime. This low number isn't surprising because the only digit out of the five that can form a prime number is 3 in the unit position. These primes are 25463, 25643, 45263, 46523, 54623 and 65423. 

The semiprimes are, not surprisingly, more numerous and they number 31. The semiprimes are 23645, 23654, 24653, 26354, 26453, 26534, 32546, 32645, 35246, 35426, 36254, 42563, 42635, 45623, 46253, 52463, 52634, 52643, 53426, 53462, 53642, 54263, 56243, 56423, 62354, 62435, 62534, 63254, 63542, 64523, 65243.

25634 is a sphenic number, meaning that it has three distinct prime factors, and so it's of particular interest to see how many of the family of 120 are sphenic. It turns out that there are 30. These are listed below but not in ascending order:

25634 = 2 * 7 * 1831

26543 = 11 * 19 * 127

26345 = 5 * 11 * 479

26435 = 5 * 17 * 311

23546 = 2 * 61 * 193

24635 = 5 * 13 * 379

24365 = 5 * 11 * 443

52346 = 2 * 7 * 3739

56234 = 2 * 31 * 907

53246 = 2 * 79 * 337

54326 = 2 * 23 * 1181

62543 = 13 * 17 * 283

62345 = 5 * 37 * 337

65342 = 2 * 37 * 883

64253 = 7 * 67 * 137

64235 = 5 * 29 * 443

32654 = 2 * 29 * 563

32465 = 5 * 43 * 151

35642 = 2 * 71 * 251

36245 = 5 * 11 * 659

34526 = 2 * 61 * 283

34562 = 2 * 11 * 1571

42653 = 13 * 17 * 193

42365 = 5 * 37 * 229

45326 = 2 * 131 * 173

45362 = 2 * 37 * 613

46235 = 5 * 7 * 1321

43265 = 5 * 17 * 509

43526 = 2 * 7 * 3109

43562 = 2 * 23 * 947

It can be seen from the above that only 43526 and 52346 share with 25634 in having 2 and 7 as distinct prime factors. However, overall there are nine permutations that have 2 and 7 as prime, but not necessarily distinct, factors. These are:

25634 = 2 * 7 * 1831

52346 = 2 * 7 * 3739

54236 = 2^2 * 7 * 13 * 149

54362 = 2 * 7 * 11 * 353

65324 = 2^2 * 7 * 2333

32564 = 2^2 * 7 * 1163

35462 = 2 * 7 * 17 * 149

43526 = 2 * 7 * 3109

43652 = 2^2 * 7 * 1559

Every sphenic number can be associated with a rectangular prism the dimensions of which correspond to the number's prime factors. In this case, the "sphenic brick" as it's sometimes called would have dimensions of 2, 7 and 1831 and an associated area of 32986 square units. This gives a volume to area ratio of about 1.28680658500429.

Unfortunately we must conclude that this family is not a happy one. A happy number has the property that repeatedly squaring the digits of the number and adding them leads to 1. However, when this process is applied to some numbers, they fall into an endless loop comprised of 4, 16, 37, 58, 89, 145, 42, 20 and they are thus not happy. All the members of this family share the same digits and, as it turns out, the process results in 90, 81, 65, 61 and 37. Thus not a single member of the family can be counted a happy number.

Drawing on another property of numbers involving their digits, D-powerful numbers can be expressed as the sum of positive powers of their digits. For example, 24536 can be expressed as \( 2^3 + 4^7 +5^3 +3^5 +6^5 \) or as \( 2^7+4^7+5+3^5+6^5 \). Not all of the other members of the family to which 24536 belongs are D-powerful. It turns out, as far as I can determine, that there are 20 D-powerful numbers amongst the family (with some having multiple representations). They are listed below with the exponents of the respective digits on the right:

24536 --> 3 7 3 5 5

24536--> 7 7 1 5 5

25346 --> 13 1 6 7 2

25436 --> 13 3 7 6 1

26354 --> 1 5 7 1 7

34256 --> 1 3 15 3 4

34256--> 5 5 15 1 3

34526 --> 4 4 3 15 4

34562 --> 6 5 1 2 15

34652 --> 6 5 1 3 15

36254 --> 4 3 15 5 3

42536 --> 7 11 5 9 4

52364 --> 5 15 4 1 7

53246 --> 1 8 3 2 6

54326 --> 5 6 8 15 5

54632 --> 5 6 6 5 9

62354 --> 2 7 10 5 2

62534 --> 3 7 5 10 2

62534--> 4 11 3 10 2

63254 --> 6 4 3 3 7

63254--> 6 4 7 1 7

63542 --> 3 10 5 5 7

63542--> 4 10 3 5 11

63542--> 4 10 5 3 3

65234 --> 2 1 11 10 6

65234--> 6 1 1 7 7

65324 --> 1 3 10 11 6

One digit-related property in which all family members share is the digital root defined as follows:

The digital root (also repeated digital sum) of a non-negative integer is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum. The process continues until a single-digit number is reached.

Because all family members share the same digits, the digital sum of all them is the same, namely 2 since the sum of the digits is 20.

Harshad numbers involve another digit-related property. These numbers, sometimes called Niven numbers, are characterised by the property that they are divisible by the sum of their digits. For every member of this family, the sum is 20 and it's thus clear that none of them can be Harshad numbers because none of them can end in the required 0. If the result of the division is a prime number then the number can be described as a Moran number and so the Moran numbers form a subset of the Harshad numbers.

Junction numbers are another class of numbers that involve the sum of a number's digits. A junction number is defined as a number that can be written as x + sod(x) for at least two x, where sod() denotes the sum of digits. It turns out that 24 members of the family are junction numbers. These are listed below with the relevant numbers in square brackets on the right:

34526 is a junction number [34498, 34507]

34625 is a junction number [34597, 34606]

35426 is a junction number [35398, 35407]

35624 is a junction number [35596, 35605]

36425 is a junction number [36397, 36406]

36524 is a junction number [36496, 36505]

43526 is a junction number [43498, 43507]

43625 is a junction number [43597, 43606]

45326 is a junction number [45298, 45307]

45623 is a junction number [45595, 45604]

46325 is a junction number [46297, 46306]

46523 is a junction number [46495, 46504]

53426 is a junction number [53398, 53407]

53624 is a junction number [53596, 53605]

54326 is a junction number [54298, 54307]

54623 is a junction number [54595, 54604]

56324 is a junction number [56296, 56305]

56423 is a junction number [56395, 56404]

63425 is a junction number [63397, 63406]

63524 is a junction number [63496, 63505]

64325 is a junction number [64297, 64306]

64523 is a junction number [64495, 64504]

65324 is a junction number [65296, 65305]

65423 is a junction number [65395, 65404]

Related to junction numbers, a self number (sometimes called a Columbian number) is a number such that there is no other number x such that x + sod(x) equals that number. However, none of the members of this family are self numbers because there is always a number x such x + sod(x) equals that number. In fact there are 96 members for which one such number exists and, as we have seen, there are 24 for which two such numbers exist. These are the junction numbers listed earlier.

A Smith number is also defined by a property involving the sum of the number's digits. It is a composite numbers with the property that the sum of its digits equals the sum of digits of its prime factors. Like the Harshad numbers mentioned earlier, none of the members of this family as Smith numbers.

Hoax numbers are similar but they only consider distinct prime factors. There are 7 members of the family that are hoax numbers, namely 23564, 24563, 32564, 36425, 45236, 64325 and 65324. We know the sum of digits of all family members is 20 and checking the distinct factors (shown in the list below), it can be seen that they two add to 20:

23564 = 2^2 * 43 * 137 (remember only count the factor 2 once)

24563 = 7 * 11^2 * 29 (remember only count the factor 11 once)

32564 = 2^2 * 7 * 1163 (remember only count the factor 2 once)

36425 = 5^2 * 31 * 47 (remember only count the factor 5 once)

45236 = 2^2 * 43 * 263 (remember only count the factor 2 once)

64325 = 5^2 * 31 * 83 (remember only count the factor 5 once)

65324 = 2^2 * 7 * 2333 (remember only count the factor 2 once)

The takeaway from this investigation is that, when exploring a family of numbers defined on the basis of the digits that comprise them, the best approach is to explore number properties that specifically involve digits. Some of these types of numbers are:

• Smith numbers
• Hoax numbers
• Harshad numbers
• Moran numbers
• Self numbers
• Junction numbers
• D-powerful numbers
• Happy numbers

ADDENDUM: today (June 18th 2019) I turned 25643 days old and this brought to mind the other family member, 25634, for which I created this post. Here is what I wrote about 25643 in my Airtable record for this number:

25643 is a Sophie Germain prime since 2 * 25643 = 51287 is also prime. 

25643 is an Ulam number, being the unique sum of two previous Ulam numbers, 69 and 25574. 

25643 is a member of OEIS A156119: primes formed by rearranging five consecutive decimal digits (avoiding leading 0). The members of this sequence, up to and including 25643, are: 10243, 12043, 20143, 20341, 20431, 23041, 24103, 25463, 25643.