

A232463


Number of ways to write n = p + q  pi(q), where p and q are odd primes not exceeding n, and pi(q) is the number of primes not exceeding q.


10



0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 1, 2, 4, 3, 3, 4, 2, 1, 2, 3, 4, 3, 2, 2, 4, 4, 4, 3, 2, 3, 6, 4, 3, 5, 2, 2, 5, 3, 4, 4, 2, 3, 5, 5, 5, 4, 2, 3, 6, 4, 4, 4, 3, 4, 6, 6, 6, 5, 2, 3, 5, 5, 7, 6, 4, 4, 5, 6, 6, 3, 3, 7, 7, 5, 4, 5, 4, 5, 6, 2, 6, 6, 4
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OFFSET

1,8


COMMENTS

Note that this sequence is different from A232443.
Conjecture: a(n) > 0 for all n > 3. Also, a(n) = 1 only for n = 4, 5, 6, 7, 9, 10, 11, 12, 15, 16, 28, 35.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588 [math.NT], 20122017.
Z.W. Sun, On a^n+ bn modulo m, arXiv preprint arXiv:1312.1166 [math.NT], 20132014.


EXAMPLE

a(10) = 1 since 10 = 7 + 7  pi(7), and 7 is an odd prime not exceeding 10.
a(11) = 1 since 11 = 5 + 11  pi(11), and 5 and 11 are odd primes not exceeding 11.
a(15) = 1 since 15 = 13 + 5  pi(5), and 13 and 5 are odd primes not exceeding 15.
a(28) = 1 since 28 = 17 + 19  pi(19), and 17 and 19 are odd primes not exceeding 28.
a(35) = 1 since 35 = 29 + 11  pi(11), and 29 and 11 are odd primes not exceeding 35.


MATHEMATICA

PQ[n_]:=n>2&&PrimeQ[n]
a[n_]:=Sum[If[PQ[nPrime[k]+k], 1, 0], {k, 2, PrimePi[n]}]
Table[a[n], {n, 1, 100}]


CROSSREFS

Cf. A000040, A000720, A002375, A232398, A232443.
Sequence in context: A240595 A083671 A201913 * A325032 A270000 A029384
Adjacent sequences: A232460 A232461 A232462 * A232464 A232465 A232466


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Nov 24 2013


STATUS

approved



