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A237348 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that prime(k) + 4 and prime(prime(m)) + 4 are both prime. 4
0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 3, 1, 2, 1, 1, 1, 2, 3, 1, 2, 2, 1, 2, 3, 3, 3, 5, 4, 2, 4, 1, 5, 1, 5, 1, 4, 4, 3, 3, 3, 1, 5, 4, 4, 3, 5, 3, 5, 6, 3, 3, 4, 3, 4, 5, 1, 5, 3, 3, 3, 5, 4, 2, 8, 1, 2, 5, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,12

COMMENTS

Conjecture: For each d = 1, 2, 3, ... there is a positive integer N(d) for which any integer n > N(d) can be written as k + m with k > 0 and m > 0 such that prime(k) + 2*d and prime(prime(m)) + 2*d are both prime. In particular, we may take (N(1), N(2), ..., N(10)) = (2, 11, 4, 15, 31, 4, 2, 77, 4, 7).

This extension of the "Super Twin Prime Conjecture" (posed by the author) implies de Polignac's well-known conjecture that any positive even number can be a difference of two primes infinitely often.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Super Twin Prime Conjecture, a message to Number Theory List, Feb. 6, 2014.

EXAMPLE

a(7) = 1 since 7 = 6 + 1 with prime(6) + 4 = 13 + 4 = 17 and prime(prime(1)) + 4 = prime(2) + 4 = 7 both prime.

a(114) = 1 since 114 = 78 + 36 with prime(78) + 4 = 397 + 4 = 401 and prime(prime(36)) + 4 = prime(151) + 4 = 877 + 4 = 881 both prime.

MATHEMATICA

pq[n_]:=pq[n]=PrimeQ[Prime[n]+4]

PQ[n_]:=PrimeQ[Prime[Prime[n]]+4]

a[n_]:=Sum[If[pq[k]&&PQ[n-k], 1, 0], {k, 1, n-1}]

Table[a[n], {n, 1, 80}]

CROSSREFS

Cf. A000040, A023200, A046132, A218829.

Sequence in context: A172069 A054348 A239660 * A037813 A159700 A083534

Adjacent sequences:  A237345 A237346 A237347 * A237349 A237350 A237351

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Feb 06 2014

STATUS

approved

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Last modified May 11 21:35 EDT 2021. Contains 343808 sequences. (Running on oeis4.)