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Old 2019-09-26, 21:03   #7
Pietro Maiorana
 
Feb 2019

3 Posts
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Each pair of odd twin primes (Oeis A077800) has an even median value.

The non-median even values ​​(complementary to Oeis A014574) are found in the sequence Oeis A100319

(Even numbers m such that at least one of m-1 and m+1 is composite)

According to my theory, A100319+1 can be divided into 5 infinite sub-sequences

a) 3*A005818; b) 5*A038179 without 2; c) 7*A007310 without 1; d) A038511; e) A025584 without 2, 3.

The first three subsequences can be rewritten respectively as:

a) 9+6*(n-1)

b) 5*(floor((41/21 - (3 mod n))^(-3*n+5)) + 3*n - 4)

c) 7/2*(6*k+(-1)^k+9)

The sequence

d) A038511 can be obtained by multiplying each term of A008364, except {1}, by itself and by all subsequent terms. Rewrite the terms in ascending order.

where A008364 is equal to

a(n) = 35n/8 + O(1). - Charles R Greathouse IV, Sep 14 2015 (see Oeis)

e) To obtain A025584 (Primes p such that p-2 is not a prime):

Write all terms of the form 2k + 7 and delete: all terms of A092256 and all terms of the form (6k + (-1) ^ k + 13)/2 [without A121764]

(Start)

A092256 (Nonprimes of form 6k+5) is the union of:

numbers of the form 5*(6k + 1), numbers of the form 7*(6k + 5), and terms of A038511 of the form 3*k+2 (End)

(Start)

To calculate A121764 (Single, or isolated or non-twin primes of form 6n + 1) consider all term of A092256+2 except those in common with:

d) A038511,
b) 5*(floor((41/21 - (3 mod n))^(-3*n+5)) + 3*n - 4),
c) 7/2*(6*k+(-1)^k+9);
(End)

QUESTION: Why did I calculate A100319 + 1 and not A100319?

ANSWER: Because some sub-sequences used in A100319 + 1 as it was possible to see in the reasoning have been re-used in the construction for a sub-sequence in A025584 (exactly Oeis A121764 Single, or isolated or non-twin primes in the form 6n + 1)

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SECOND METHOD
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Consider only the sequence A025584 (Primes p such that p-2 is not a prime):

To obtain A025584 (without 2, 3)
Write all terms of the form 2k + 7 and delete: all terms of A092256 and all terms of the form (6k + (-1) ^ k + 13)/2 [without A121764]

(Start)

A092256 (Nonprimes of form 6k+5) is the union of:


numbers of the form 5*(6k + 1),
numbers of the form 7*(6k + 5),
and terms of A038511 of the form 3*k+2
(End)

(Start)

To calculate A121764 (Single, or isolated or non-twin primes of form 6n + 1)
consider all term of A092256+2 except those in common with:


a) A038511,
b) 5*(floor((41/21 - (3 mod n))^(-3*n+5)) + 3*n - 4),
c) 7/2*(6*k+(-1)^k+9);
(End)

While A038511 can be obtained by multiplying each term of 35n/8 + O(1) , except {1}, by itself and by all subsequent terms. Rewrite the terms in ascending order.

Well, if you remove from the infinite list of prime numbers, the values of A025584 can be obtained A006512 "Greater of twin primes"
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