login
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
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
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588 [math.NT], 2012-2017.
Z.-W. Sun, On a^n+ bn modulo m, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.
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[n-Prime[k]+k], 1, 0], {k, 2, PrimePi[n]}]
Table[a[n], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Nov 24 2013
STATUS
approved