Sums of distinct prime factors

(Dall.e creation)

We add two integers, 

we list the distinct prime factors of the result,

we make the sum S of those factors.

Example

As we are in December 2023, let's consider the pair {12;2023}.

12 + 2023 = 2035

The distinct prime factors of 2035 are 5, 11 and 37.

Their sum is S = 5 + 11 + 37 = 53.

We will now propose a series of sequences related to the nature of this sum S (many thanks to Giorgos Kalogeropoulos for the computed terms and graphs!-)

A] Lexicographically earliest sequence of distinct positive integers such that the sum of the distinct prime factors of a(n) + a(n + 1) is a prime.

A = 1, 2, 3, 4, 5, 6, 7, 9, 8, 10, 12, 11, 13, 14, 15, 16, 18, 19, 17, 20, 21, 22, 25, 23, 24, 26, 27, 31, 28, 30, 29, 32, 35, 33, 34, 37, 36, 43, 38, 41, 39, 40, 42, 46, 50, 47, 49, 48, 52, 44, 45, 51, 56, 53, 54, 55, 58, 60, 61, 57, 59, 62, 63, 64, 67, 69, 68, 71, 65, 66, 70, 72, 77, 74, 75, 76, 73, 78, 79, 81, 82, 80, 83, 84, 85, 88, 91, 90, 86, 87, 89, 92, 99, 93, 98, 94, 97, 95, 96, 101, ...

B] Lexicographically earliest sequence of distinct positive integers such that the sum of the distinct prime factors of a(n) + a(n + 1) is a nonprime.

B = 1, 13, 2, 12, 3, 11, 4, 10, 5, 9, 6, 8, 7, 14, 16, 17, 18, 15, 20, 19, 23, 22, 24, 21, 25, 26, 29, 27, 28, 32, 30, 33, 36, 34, 31, 35, 39, 37, 38, 40, 44, 41, 43, 42, 45, 46, 47, 48, 50, 49, 53, 51, 54, 52, 58, 56, 55, 57, 60, 59, 61, 62, 64, 65, 67, 63, 66, 68, 70, 71, 69, 72, 73, 74, 76, 77, 75, 78, 80, 79, 82, 84, 86, 85, 81, 87, 83, 88, 89, 91, 92, 90, 93, 94, 95, 99, 96, 98, 97, 101, ...

C] Lexicographically earliest sequence of distinct positive integers such that the sum of the distinct prime factors of a(n) + a(n + 1) is a nonnegative even number.

C = 1, 3, 5, 10, 6, 2, 13, 8, 7, 9, 12, 4, 11, 19, 14, 16, 17, 15, 18, 21, 24, 27, 28, 23, 22, 20, 25, 26, 29, 31, 32, 33, 30, 34, 35, 40, 37, 38, 39, 36, 41, 43, 42, 45, 46, 44, 47, 48, 51, 59, 52, 50, 49, 53, 57, 54, 56, 55, 60, 63, 65, 58, 61, 62, 64, 66, 67, 68, 70, 71, 69, 72, 73, 74, 76, 77, 78, 75, 79, 80, 81, 87, 83, 85, 86, 82, 88, 89, 91, 84, 90, 92, 93, 94, 95, 103, 98, 100, 101, 97, ...

D] Lexicographically earliest sequence of distinct positive integers such that the sum of the distinct prime factors of a(n) + a(n + 1) is an odd number.

D = 1, 2, 3, 4, 5, 6, 7, 10, 8, 9, 11, 12, 13, 14, 15, 16, 18, 19, 17, 20, 21, 22, 24, 23, 25, 27, 26, 28, 30, 29, 32, 35, 33, 34, 37, 31, 36, 38, 41, 39, 40, 42, 44, 45, 43, 46, 48, 49, 47, 50, 51, 52, 53, 54, 55, 57, 56, 60, 58, 63, 59, 62, 65, 66, 61, 64, 67, 69, 68, 71, 73, 75, 74, 70, 72, 76, 81, 77, 80, 78, 79, 83, 82, 84, 85, 87, 86, 90, 88, 91, 93, 95, 89, 92, 96, 97, 94, 98, 99, 100, ...

E] Lexicographically earliest sequence of distinct positive integers such that the sum of the distinct prime factors of a(n) + a(n + 1) is a Fibonacci number.

E = 1, 2, 3, 5, 4, 8, 7, 6, 9, 13, 11, 14, 10, 12, 15, 17, 19, 25, 20, 16, 22, 23, 21, 24, 30, 18, 26, 28, 36, 39, 33, 31, 41, 34, 38, 37, 27, 45, 43, 29, 35, 40, 32, 44, 49, 47, 42, 46, 50, 56, 52, 54, 71, 57, 51, 55, 53, 72, 63, 62, 66, 59, 69, 75, 60, 48, 58, 67, 61, 64, 80, 65, 70, 74, 78, 84, 68, 76, 86, 83, 79, 73, 89, 85, 77, 92, 82, 87, 105, 90, 102, 93, 81, 88, 104, 91, 101, 94, 98, 97, ...

F] Lexicographically earliest sequence of distinct positive integers such that the sum of the distinct prime factors of a(n) + a(n + 1) is a palindome in base 10.

F = 1, 2, 3, 4, 5, 6, 8, 7, 9, 11, 13, 12, 15, 10, 14, 18, 22, 23, 17, 19, 21, 24, 16, 20, 25, 29, 27, 30, 26, 28, 34, 38, 37, 35, 40, 32, 43, 42, 33, 31, 41, 39, 36, 44, 52, 46, 50, 48, 53, 45, 51, 47, 49, 57, 55, 66, 58, 54, 67, 56, 65, 59, 62, 61, 60, 63, 68, 76, 75, 69, 82, 78, 73, 71, 64, 80, 91, 90, 70, 74, 77, 83, 79, 72, 88, 93, 89, 92, 99, 95, 86, 85, 96, 98, 84, 87, 94, 97, 103, 101, ...

Update, a few hours later, thanks to Giorgos K.

> GK

> I was looking for some variations for the sum S of distinct prime factors, trying to avoid the straight-line-graphs:

> G] Lexicographically earliest sequence of distinct positive integers such that n divides the sum of the distinct prime factors (sopf) of a(n - 1) + a(n).

> G = 1, 3, 6, 9, 12, 23, 10, 5, 51, 39, 18, 17, 27, 43, 61, 26, 67, 11, 8, 83, 69, 16, 7, 88, 4, 65, 93, 22, 243, 128, 49, 38, 24, 121, 80, 75, 138, 79, 217, 102, 135, 50, 32, 91, 81, 48, 230, 28, 66, 225, 77, 158, 151, 178, 34, 125, 420, 97, 468, 62, 56, 475, 13, 170, 211, 94, 299, 311, 604, 45, 372, 30, 112, 101, 191, 164, 289, 76, 863, 131, 185, 52, 31, 364, 137, 416, 450, 546, 319, 262, 275, 268, 286, 159, 407, 216, 357, 222, 166, 425, ...

[This will be soon here]

> H] Lexicographically earliest sequence of distinct positive integers such that the sum of the distinct prime factors (sopf) of a(n) + a(n + 1) is a perfect square.

> H = 1, 13, 15, 24, 4, 10, 18, 21, 7, 32, 14, 25, 3, 11, 17, 22, 6, 8, 20, 19, 9, 5, 23, 16, 12, 2, 26, 29, 27, 28, 38, 54, 40, 52, 42, 50, 44, 48, 46, 66, 51, 41, 53, 39, 55, 37, 57, 35, 31, 61, 33, 59, 58, 34, 60, 72, 45, 47, 65, 67, 88, 70, 62, 30, 36, 56, 76, 79, 104, 80, 75, 83, 49, 43, 69, 63, 92, 91, 64, 68, 87, 71, 84, 74, 81, 77, 78, 105, 93, 90, 94, 89, 95, 101, 82, 73, 85, 98, 86, 97, ...

[This will be soon here]

> I] Lexicographically earliest sequence of distinct positive integers such that the sum of the distinct prime factors (sopf) of a(n) + a(n + 1) is a perfect cube.

> I = 1, 14, 31, 44, 91, 92, 43, 2, 13, 32, 103, 80, 55, 20, 25, 50, 85, 98, 37, 8, 7, 38, 97, 86, 49, 26, 19, 56, 79, 104, 121, 62, 73, 110, 115, 68, 67, 116, 109, 74, 61, 122, 163, 132, 3, 12, 33, 42, 93, 90, 45, 30, 15, 60, 75, 108, 27, 18, 57, 78, 105, 120, 63, 72, 111, 24, 21, 54, 81, 102, 123, 162, 133, 152, 143, 40, 5, 10, 35, 100, 83, 52, 23, 22, 53, 82, 101, 34, 11, 4, 41, 94, 89, 46, 29, 16, 59, 76, 107, 28, ...

[This will be soon here]

The last two seqs have marvelous graphs, indeed, Giorgos! Bravo and thanks for the good job!