Some Sequences Involving Prime Factors

One of the properties of the number associated with my diurnal age today, 27335, is that it's a member of OEIS A020700:

A020700

Numbers \(k\) such that \(k\) + sum of its prime factors = (\(k\)+1) + sum of its prime factors.

Let's confirm that 27335 does indeed have this property:$$ \begin{align} 27335 &= 5 \times 7 \times 11 \times 71 \\5 + 7 + 11 + 71 &= 94\\27335+94 &=27429 \\27336 &= 2^3 \times 3 \times 17 \times 67\\ 2+2+2 + 3 + 17+ 67 &= 93\\ 27336+93 &= 27429 \end{align} $$The initial members of the sequence are (permalink):

7, 14, 63, 80, 224, 285, 351, 363, 475, 860, 902, 1088, 1479, 2013, 2023, 3478, 3689, 3925, 5984, 6715, 8493, 9456, 13224, 15520, 17227, 18569, 19502, 20490, 21804, 24435, 24476, 27335, 31899, 32390, 35815, 37406, 37582, 41876, 49468, 50609, 54137, 57239

Another way to phrase the property of members of this sequence is to say that the addition of the product and sum of the prime factors of \(n\) is equal to the addition of the product and sum of the prime factors of \(n+1\). Let's take 63 as an example:$$ 63 = 3 \times 3 \times 7 \\ \underbrace{3 \times 3 \times 7}_{\text{product of prime factors}} + \underbrace{3 + 3 +7}_{\text{sum of prime factors}} = 76 \\ \text{and}\\64 =2 \times 2 \times 2 \times 2 \times 2 \times 2 \\ \underbrace{2 \times 2 \times 2 \times 2 \times 2 \times 2}_{\text{product of prime factors}}+ \underbrace{2+2+2+2+2+2}_{\text{sum of prime factors}} = 76 $$This got me thinking about pairs of consecutive numbers that have the same sum of their distinct prime factors. To my surprise, this led me to the famous Ruth-Aaron numbers that I looked at in a post titled Ruth-Aaron Pairs and eRAPs on Monday the 29th of April 2019. The numbers are so called because 714 is Babe Ruth's lifetime home run record and Hank Aaron's 715th home run broke this record. 714 and 715 have the same sum of prime divisors, taken with multiplicity. These numbers constitute OEIS A039752:

A039752

Ruth-Aaron numbers (2): sum of prime divisors of \(n\) = sum of prime divisors of \(n\)+1 (both taken with multiplicity).

The initial members are (permalink):

5, 8, 15, 77, 125, 714, 948, 1330, 1520, 1862, 2491, 3248, 4185, 4191, 5405, 5560, 5959, 6867, 8280, 8463, 10647, 12351, 14587, 16932, 17080, 18490, 20450, 24895, 26642, 26649, 28448, 28809, 33019, 37828, 37881, 41261, 42624, 43215, 44831, 44891, 47544, 49240

If multiplicity is ignored and only distinct prime factors are considered, then we have OEIS A006145:

A006145

Ruth-Aaron numbers (1): sum of prime divisors of \(n\) = sum of prime divisors of \(n\)+1.

The initial members are (permalink):

5, 24, 49, 77, 104, 153, 369, 492, 714, 1682, 2107, 2299, 2600, 2783, 5405, 6556, 6811, 8855, 9800, 12726, 13775, 18655, 21183, 24024, 24432, 24880, 25839, 26642, 35456, 40081, 43680, 48203, 48762, 52554, 61760, 63665, 64232, 75140, 79118, 95709, 106893, 109939

Looking the 714 and 715 it can be seen that:$$ \begin{align} 714 &= 2 \times 3 \times 7 \times 17 \\ 2+3+7+17 &= 29 \\715 &= 5 \times 11 \times 13\\ 5+11+13 &= 29 \end{align}$$The two sequences will share terms that are square-free (like 714) but otherwise differ. For example, the second term of the previous sequence is 24 with distinct prime factors of 2 and 3, adding to 5. The next term 25 has only one distinct prime factor, 5, and thus 24 qualifies for inclusion in the sequence but it is not a member of OEIS  A039752.

In my blog post Ruth-Aaron Pairs and eRAPs, mentioned earlier, the eRAP refers to an extension, proposed by Abhiram R. Devesh, of Ruth-Aaron Pairs (thus called eRAP) where two consecutive numbers form a pair if the sums of their prime factors are consecutive. 

These numbers form OEIS  A228126:

A228126

Sum of prime divisors of \(n\) (with repetition) is one less than the sum of prime divisors (with repetition) of \(n\)+1.

The initial members are:

2, 3, 4, 9, 20, 24, 98, 170, 1104, 1274, 2079, 2255, 3438, 4233, 4345, 4716, 5368, 7105, 7625, 10620, 13350, 13775, 14905, 20220, 21385, 23408, 25592, 26123, 28518, 30457, 34945, 35167, 38180, 45548, 49230, 51911, 52206, 53456, 56563, 61456, 65429, 66585

For example, let's take 24:$$ \begin{align} 24 &= 2 \times 2 \times 2 \times 3 \\2+2+2+3& =9\\25 &=5 \times 5\\5+5&=10 \end{align}$$Since 10 is one more than 9, 24 qualifies for membership in the sequence.

In my earlier post, I spend some time looking at triples comprising three consecutive numbers as well as pairs comprising two consecutive numbers and so it's worth looking back at that post for details about that. Referring back to OEIS A020700, it's likely that triples exist there as well but they would be rather be comprised of quite large large numbers. It can be noted also that all of these sequences are base-independent.