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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, Table of n, a(n) for n = 1..10000

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

Cf. A000040, A000217, A000290, A000720, A237598, A237840, A238504, A239884.

Sequence in context: A005793 A183391 A029346 * A030496 A005794 A280860

Adjacent sequences:  A238899 A238900 A238901 * A238903 A238904 A238905

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Mar 06 2014

STATUS

approved

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Last modified January 18 13:09 EST 2019. Contains 319271 sequences. (Running on oeis4.)