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A289827 a(n) = largest m <= n such that pi(m + n) = pi(m) + pi(n), where pi function is A000720 (with pi(0) = 0). 1
0, 2, 2, 4, 2, 2, 1, 1, 4, 10, 2, 2, 1, 1, 4, 4, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 4, 4, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 4, 4, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 4, 4, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 4, 4, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 10, 10 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

It seems that the sequence is bounded, namely a(n) <= 10.

We have a(n) = 10 for n = 10, 99, 100, 189, 190, 819, 820, ...

For n > 9; a(n) = a(n+1) = 10 if and only if n+2 is in A007530.

First conjecture: for n > 1, all a(n) belong to the set {1, 2, 4, 10}.

Second Hardy-Littlewood conjecture: pi(x+y) <= pi(x) + pi(y) for x,y >= 2.

Third conjecture (T. Ordowski): pi(x+y) < pi(x) + pi(y) for x,y >= 11.

Carl Pomerance (in a letter to the author) wrote: I believe if correct, your conjecture would disprove the Hardy-Littlewood prime k-tuples conjecture, as shown by Hensley and Richards over 30 years ago. They showed that prime k-tuples implies that there are pairs y < x with pi(x+y) >= pi(x) + pi(y) and pi(y) arbitrarily large. Since pi(2x) < 2*pi(x), by increasing y in a y,x example, one would come on a new pair y' < x with pi(x+y') = pi(x) + pi(y'). - Thomas Ordowski, Aug 14 2017

By the k-tuple conjecture, the smallest a(n) > 10 is 1418 for some n > 10^100. - Nathan McNew, Aug 17 2017

a(n) > 0 for n > 1.

LINKS

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

Douglas Hensley and Ian Richards, Primes in Intervals. Acta Mathematica 25,4 (1973/1974) 375-391.

Ian Richards, On the Incompatibility of Two Conjectures Concerning Primes;..., Bull. Amer. Math. Soc. 80,3 (1974) 419-438.

Eric Weisstein's MathWorld, Hardy-Littlewood conjectures

Wikipedia, Second Hardy-Littlewood conjecture

MAPLE

f:= proc(n) local m;

  for m from n by -1 do

    if numtheory:-pi(m+n)=numtheory:-pi(m)+numtheory:-pi(n)

        then return m

      fi

  od

end proc:

map(f, [$1..100]); # Robert Israel, Aug 14 2017

MATHEMATICA

Table[SelectFirst[Range[n, 0, -1], PrimePi[# + n] == PrimePi[#] + PrimePi[n] &], {n, 100}] (* Michael De Vlieger, Aug 15 2017 *)

f[n_] := Block[{m = n, p = PrimePi@ n}, While[ PrimePi[m + n] != PrimePi[m] + p, m--]; m]; Array[f, 103] (* Robert G. Wilson v, Aug 30 2017 *)

PROG

(PARI) a(n) = my(m=n); while(1, if(primepi(m+n)==primepi(m)+primepi(n), return(m)); m--) \\ Felix Fröhlich, Aug 13 2017

CROSSREFS

Cf. A000720, A007530.

Sequence in context: A261872 A021450 A239675 * A092188 A097884 A094818

Adjacent sequences:  A289824 A289825 A289826 * A289828 A289829 A289830

KEYWORD

nonn

AUTHOR

Thomas Ordowski, Aug 13 2017

EXTENSIONS

More terms from Altug Alkan and Robert Israel, Aug 13 2017

STATUS

approved

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Last modified May 22 18:53 EDT 2019. Contains 323481 sequences. (Running on oeis4.)