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!)
A238902 a(n) = |{0 < k <= n: pi(pi(k*n)) is a square}|, where pi(x) denotes the number of primes not exceeding x. 3
1, 2, 1, 1, 2, 3, 2, 1, 2, 4, 3, 4, 3, 3, 3, 2, 5, 5, 4, 3, 5, 4, 5, 4, 5, 5, 6, 4, 4, 6, 4, 5, 4, 6, 4, 4, 3, 4, 4, 3, 4, 4, 4, 4, 5, 3, 4, 5, 4, 3, 4, 5, 5, 4, 2, 2, 3, 2, 3, 3, 3, 1, 4, 3, 4, 3, 3, 3, 5, 2, 1, 2, 3, 5, 3, 4, 4, 2, 1, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Conjecture: (i) a(n) > 0 for all n > 0.
(ii) For every n = 1, 2, 3, ..., there exists a positive integer k <= (n+1)/2 such that pi(pi(k*n)) is a triangular number.
We have verified parts (i) and (ii) for n up to 2*10^5 and 10^5 respectively.
See A239884 for a sequence related to part (i) of the conjecture.
LINKS
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.
EXAMPLE
a(8) = 1 since pi(pi(3*8)) = pi(pi(24)) = pi(9) = 2^2.
a(434) = 1 since pi(pi(297*434)) = pi(pi(128898)) = pi(12064) = 38^2.
a(1042) = 1 since pi(pi(698*1042)) = pi(pi(727316)) = pi(58590) = 77^2.
a(9143) = 1 since pi(pi(8514*9143)) = pi(pi(77843502)) = pi(4550901) = 565^2.
a(48044) > 0 since pi(pi(18332*48044)) = pi(45075237) = 1650^2.
a(52158) > 0 since pi(pi(27976*52158)) = pi(72792062) = 2067^2.
a(78563) > 0 since pi(pi(26031*78563)) = pi(100326489) = 2404^2.
a(98213) > 0 since pi(pi(37308*98213)) = pi(174740922) = 3123^2.
a(141589) > 0 since pi(pi(42375*141589)) = pi(279538049)= 3899^2.
a(154473) > 0 since pi(pi(42954*154473)) = pi(307695484) = 4080^2.
a(195387) > 0 since pi(pi(60161*195387)) = pi(530982180) = 5282^2.
MATHEMATICA
SQ[n_]:=IntegerQ[Sqrt[n]]
p[k_, n_]:=SQ[PrimePi[PrimePi[k*n]]]
a[n_]:=Sum[If[p[k, n], 1, 0], {k, 1, n}]
Table[a[n], {n, 1, 80}]
PROG
(PARI) {a(n) = sum( k=1, n, issquare( primepi( primepi( k*n))))}; /* Michael Somos, Mar 10 2014 */
CROSSREFS
Sequence in context: A005793 A183391 A029346 * A329750 A030496 A005794
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Mar 06 2014
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 March 28 05:02 EDT 2024. Contains 371235 sequences. (Running on oeis4.)