login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A234567 Number of ways to write n = k + m with k > 0 and m > 0 such that p = phi(k) + phi(m)/2 + 1 and P(p-1) are both prime, where phi(.) is Euler's totient function and P(.) is the partition function (A000041). 14
0, 0, 0, 1, 2, 1, 1, 3, 2, 2, 3, 2, 4, 2, 4, 4, 2, 4, 3, 5, 1, 3, 2, 3, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 2, 0, 1, 2, 1, 1, 2, 1, 2, 3, 2, 8, 2, 1, 2, 2, 3, 3, 1, 2, 7, 0, 2, 3, 3, 4, 5, 7, 3, 4, 1, 9, 1, 4, 3, 1, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
Conjecture: (i) a(n) > 0 for all n > 727.
(ii) For the strict partition function q(.) (cf. A000009), any n > 93 can be written as k + m with k > 0 and m > 0 such that p = phi(k) + phi(m)/2 + 1 and q(p-1) - 1 are both prime.
(iii) If n > 75 is not equal to 391, then n can be written as k + m with k > 0 and m > 0 such that f(k,m) - 1, f(k,m) + 1 and q(f(k,m)) + 1 are all prime, where f(k,m) = phi(k) + phi(m)/2.
Part (i) of the conjecture implies that there are infinitely many primes p with P(p-1) prime.
LINKS
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014
EXAMPLE
a(21) = 1 since 21 = 6 + 15 with phi(6) + phi(15)/2 + 1 = 7 and P(6) = 11 both prime.
a(700) = 1 since 700 = 247 + 453 with phi(247) + phi(453)/2 + 1 = 367 and P(366) = 790738119649411319 both prime.
a(945) = 1 since 945 = 687 + 258 with phi(687) + phi(258)/2 + 1 = 499 and P(498) = 2058791472042884901563 both prime.
MATHEMATICA
f[n_, k_]:=EulerPhi[k]+EulerPhi[n-k]/2
q[n_, k_]:=PrimeQ[f[n, k]+1]&&PrimeQ[PartitionsP[f[n, k]]]
a[n_]:=Sum[If[q[n, k], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 100}]
CROSSREFS
Sequence in context: A060118 A329143 A219032 * A241950 A316776 A029308
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Dec 28 2013
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 15 12:56 EDT 2024. Contains 374332 sequences. (Running on oeis4.)