|
|
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
|
Douglas Hensley and Ian Richards, Primes in Intervals. Acta Mathematica 25,4 (1973/1974) 375-391.
|
|
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:
|
|
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
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|