

A108309


Consider the triangle of odd numbers where the nth row has the next n odd numbers. The sequence is the number of primes in the nth row.


5



0, 2, 2, 3, 2, 3, 3, 4, 4, 5, 3, 4, 6, 4, 6, 6, 4, 6, 7, 6, 8, 7, 5, 8, 9, 8, 7, 8, 9, 8, 9, 10, 10, 8, 10, 12, 5, 12, 12, 13, 9, 11, 11, 9, 13, 14, 9, 14, 14, 10, 10, 19, 14, 12, 12, 12, 12, 16, 15, 16, 15, 13, 18, 16, 16, 12, 16, 17, 15, 16, 18, 14, 15, 20, 18, 19, 14, 19, 20, 18, 16
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OFFSET

1,2


COMMENTS

Except for the initial term, a(n)>=2 because in the interval 2n1 of odd numbers there are always at least two primes.
For n>2, this is the same as the number of primes between n^2n and n^2+n, which is the sum of A089610 and A094189.  T. D. Noe, Sep 16 2008
a(n) = SUM(A010051(A176271(n,k)): 1<=k<=n).  Reinhard Zumkeller, Apr 13 2010
From Pierre CAMI, Sep 03 2014: (Start)
For n>1 a(n)~floor(1/2 + n/log(n)).
The number of primes < n^2 is ~ n^2/2/log(n) by the prime number theorem, as a(n) ~ floor(1/2 + n/log(n)) we have:
n^2/2/log(n) ~ 1 + floor(1/2 + 2/log(2)) + floor(1/2 + 3/log(3)) + floor(1/2 + 4/log(4)) + ... + floor(1/2 + (n1)/log(n1)) + floor(1/2 + n/log(n)).
For n=16000 the number of primes < n^2 is 13991985, the sum: 1 + floor(1/2 + 2/log(2)) + floor(1/2 + 3/log(3)) + floor(1/2 + 4/log(4))+ ... + floor(1/2 + (n1)/log(n1)) + floor(1/2 + n/log(n)) is 13991101 and (n^2)/(2*log(n)) is 13222671.
So between n^2+n and n^2+3*n there are n odd numbers and ~floor(1/2 + n/log(n)) prime numbers.
The twin primes are of the form T1=n^2+n1 and T2=n^2+n+1, or n^2+n+T1 and n^2+n+T2 with T1<=2*n1, or n^2+n+P and n^2+n+P(2 or +2) with P prime <=2*n1.
(End)


LINKS

T. D. Noe, Table of n, a(n) for n=1..10000
Pierre CAMI, Table of n, a(n) and Floor(1/2+n/log(n))for n=1..10000


EXAMPLE

Triangle begins:
1: 1 > 0 primes,
2: 3,5 > 2 primes,
3: 7,9,11 > 2 primes,
4: 13,15,17,19 > 3 primes.


MAPLE

seq(numtheory:pi(n^2+n1)numtheory:pi(n^2n), n=1..100); # Robert Israel, Sep 03 2014


MATHEMATICA

f[n_] := PrimePi[n^2 + n  1]  PrimePi[n^2  n]; Table[f[n], {n, 81}] (* Ray Chandler, Jul 26 2005 *)


PROG

(Haskell)
a108309 = sum . (map a010051) . a176271_row
 Reinhard Zumkeller, May 24 2012


CROSSREFS

Cf. A014085, A088485.
Sequence in context: A029213 A029209 A282630 * A103469 A029326 A239500
Adjacent sequences: A108306 A108307 A108308 * A108310 A108311 A108312


KEYWORD

easy,nonn


AUTHOR

Giovanni Teofilatto, Jul 25 2005


EXTENSIONS

Edited and extended by Ray Chandler, Jul 26 2005


STATUS

approved



