A238902 a(n) = |{0 < k <= n: pi(pi(k*n)) is a square}|, where pi(x) denotes the number of primes not exceeding x.
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
Offset: 1
Keywords
Examples
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.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
- Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.
Programs
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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}] -
PARI
{a(n) = sum( k=1, n, issquare( primepi( primepi( k*n))))}; /* Michael Somos, Mar 10 2014 */
Comments