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a(n) is the maximal difference between the corresponding terms of sequences defined in the same way as A159559, but with initial terms A001359(n-1)+2 and A001359(n-1) respectively.
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4
4, 14, 6, 6, 6, 12, 6, 8, 14, 14, 18, 36, 24, 65, 18, 6, 10, 6, 84, 14, 162, 10, 54, 84, 179, 10, 23, 12, 18, 18, 24, 128, 18, 24, 28, 10, 10, 72, 34, 23, 12, 18, 6, 6, 12, 34, 8, 644, 12, 12, 6, 29, 24, 12, 18, 28, 28, 24, 22, 22, 10, 14, 12, 12, 16, 6, 58
COMMENTS
It seems likely that 6 occurs infinitely often.
EXAMPLE
Since A276703(3)=4 (cf. example there), a(2)=4.
For a lesser p of twin primes, let B_k be A159559, but with initial term k; then a(n) is the smallest m such that B_(p+2)(m)-B_p(m)>6, where p = A001359(n-1), or a(n) = 0 if there is no such m.
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4
0, 13, 0, 0, 0, 9, 0, 11, 11, 5, 3, 15, 3, 7, 3, 0, 3, 0, 3, 5, 7, 3, 11, 5, 3, 5, 11, 3, 9, 3, 3, 7, 3, 5, 5, 3, 5, 3, 5, 11, 3, 5, 0, 0, 5, 5, 7, 5, 13, 7, 0, 5, 3, 3, 3, 3, 7, 3, 3, 3, 5, 3, 7, 3, 3, 0, 3, 5, 5, 3, 11, 11, 5, 3, 5, 7, 5, 3, 0, 3, 3, 3, 3, 3
COMMENTS
Theorem: a(n) takes only the values 0, 3, 5, 7, 9, 11, 13, 15, and 17.
PROG
(PARI) nextcomposite(n)=if(n<4, return(4)); n=ceil(n); if(isprime(n), n+1, n)
do(p)=my(a=p, b=p+2, f); for(n=3, 17, f=if(isprime(n), nextprime, nextcomposite); a=f(a+1); b=f(b+1); if(b-a > 6, return(n))); 0
For a lesser p of twin primes, let B_(p+2) and B_p be sequences defined as A159559, but with initial terms p+2 and p respectively. The sequence lists p for which all differences B_(p+2)(n)-B_p(n)<=6.
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3
3, 11, 17, 29, 59, 227, 269, 1277, 1289, 1607, 2129, 2789, 3527, 3917, 4637, 4787, 5639, 8999, 13679, 14549, 18119, 27737, 36779, 38447, 39227, 44267, 62129, 71327, 75989, 80669, 83219, 88799, 93479, 97367, 99707, 113147, 113159, 115769, 122027, 122387, 124337, 124769, 132749, 150209, 160079
COMMENTS
B_(p+2)(n) - B_p(n) < 6 for all n >= 2 if and only if p = 3.
It is astonishing that, although terms a(n) == 7 or 9 (mod 10) occur often, the first terms a(n)==1 (mod 10) are 11, 165701, ... (cf. A022009). This phenomenon is explained in the Shevelev link.
PROG
(PARI) nextcomposite(n)=if(n<4, return(4)); n=ceil(n); if(isprime(n), n+1, n)
is(n)=if(!isprime(n) || !isprime(n+2), return(0)); my(p=n, q=n+2, k=2, f); while(p!=q && q-p<7, f=if(isprime(k++), nextprime, nextcomposite); p=f(p+1); q=f(q+1)); p==q \\ Charles R Greathouse IV, Sep 21 2016
For a lesser p= A001359(n-1), n>=2, of twin primes, let B_k be the sequence defined as A159559 but with initial term k; a(n) is the smallest m such that B_(p+2)(m)-B_p(m) = max_{t>=2} (B_(p+2)(t)-B_p(t)).
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2
5, 17, 11, 5, 3, 17, 3, 11, 11, 5, 31, 107, 13, 333, 17, 5, 3, 3, 281, 5, 997, 3, 487, 659, 5178, 5, 15, 3, 23, 53, 13, 1567, 13, 13, 181, 3, 5, 443, 37, 21, 19, 11, 5, 3, 5, 5, 7, 20786, 13, 7, 5, 21, 3, 5, 17, 61, 31, 23, 7, 3, 11, 5, 11, 5, 3, 3, 157, 37
FORMULA
B_(p+2)(a(n)) - B_p(a(n)) = A276826(n).
EXAMPLE
Let n=2, p= A001359(1)=3. Then B_3(2)=3, B_3(3)=5, B_3(4)=6, B_3(5)=7, B_3(6)=8, B_3(7)=11, B_3(8)=12, B_3(9)=14, B_3(10)=15, B_3(11)=17; Further, B_5(2)=5, B_5(3)=7, B_5(4)=8, B_5(5)=11, B_5(6)=12, B_5(7)=13, B_5(8)=14, B_5(9)=15, B_5(10)=16, B_5(11)=17 and, beginning with t=11,
B_3 merges with B_5. So, max(B_5(t)-B_3(t))=4 reaching at t=5 and t=6.
Thus a(2)=min(5,6)=5.
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