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A237496 Number of ordered ways to write n = k + m (0 < k <= m) with pi(k) + pi(m) - 2 prime, where pi(.) is given by A000720. 6
0, 0, 0, 0, 0, 1, 2, 4, 4, 3, 2, 3, 3, 3, 5, 3, 1, 4, 5, 5, 7, 4, 1, 2, 1, 1, 1, 1, 1, 3, 6, 7, 8, 8, 8, 8, 8, 9, 11, 11, 11, 11, 9, 7, 7, 4, 1, 2, 1, 2, 3, 5, 7, 10, 14, 14, 14, 10, 6, 10, 14, 16, 19, 16, 13, 12, 11, 10, 7, 6, 5, 3, 3, 4, 3, 6, 9, 13, 17, 18 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

Conjecture: (i) a(n) > 0 for all n > 5.

(ii) Any integer n > 23 can be written as k + m (k > 0 and m > 0) with pi(k) + pi(m) prime. Also, each integer n > 25 can be written as k + m (k > 0 and m > 0) with pi(k) + pi(m) - 1 prime.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..3000

EXAMPLE

a(6) = 1 since 6 = 3 + 3 with pi(3) + pi(3) - 2 = 2 + 2 - 2 = 2 prime.

a(17) = 1 since 17 = 2 + 15 with pi(2) + pi(15) - 2 = 1 + 6 - 2 = 5 prime.

a(99) = 1 since 99 = 1 + 98 with pi(1) + pi(98) - 2 = 0 + 25 - 2 = 23 prime.

MATHEMATICA

PQ[n_]:=n>0&&PrimeQ[n]

p[k_, m_]:=PQ[PrimePi[k]+PrimePi[m]-2]

a[n_]:=Sum[If[p[k, n-k], 1, 0], {k, 1, n/2}]

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

CROSSREFS

Cf. A000040, A000720, A232465, A237284, A237291, A237453, A237497.

Sequence in context: A185356 A008777 A306691 * A202690 A257978 A193358

Adjacent sequences:  A237493 A237494 A237495 * A237497 A237498 A237499

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Feb 08 2014

STATUS

approved

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Last modified July 11 18:00 EDT 2020. Contains 335645 sequences. (Running on oeis4.)