A278114 -id:A278114 - cp's OEIS frontend

A278115 Triangle T(n,k) = A278113(n,k)^2 A000040(k) for 1 <= k <= A278114(n), read by rows.

2, 8, 3, 5, 7, 18, 12, 5, 7, 11, 13, 17, 32, 27, 20, 28, 11, 13, 17, 19, 23, 29, 31, 50, 48, 45, 28, 44, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 72, 48, 45, 63, 44, 52, 68, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 98, 75, 80, 63, 44, 52, 68, 76, 92, 29, 31, 37, 41, 43, 47, 53

Offset: 1

Jason Kimberley, Feb 10 2017

			The first six rows are:
2;
8, 3, 5, 7;
18, 12, 5, 7, 11, 13, 17;
32, 27, 20, 28, 11, 13, 17, 19, 23, 29, 31;
50, 48, 45, 28, 44, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47;
72, 48, 45, 63, 44, 52, 68, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71;

Jason Kimberley, Table of i, a(i) for i = 1..10126 (T(n,k) for n = 1..46)

Cf. A278101.

A278112:=func;
A278115_row:=funcA278112(n,p)^2*p:p in PrimesUpTo(2*n^2)]>;
&cat[A278115_row(n):n in[1..7]];

Table[# Floor[n Sqrt[2/#]]^2 &@ Prime@ k, {n, 7}, {k, PrimePi[2 n^2]}] // Flatten (* Michael De Vlieger, Feb 17 2017 *)

T(n,k) = prime(k) * floor(n*sqrt(2/prime(k)))^2.

A278113 Triangle T(n,k) = A278112(n,A000040(k)) for 1 <= k <= A278114(n), read by rows.

Original entry on oeis.org

1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 5, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 5, 4, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 6, 5, 4, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Jason Kimberley, Feb 09 2017

This triangle consists of those columns of A278112 that have prime index.

			The first eight rows are:
  1;
  2, 1, 1, 1;
  3, 2, 1, 1, 1, 1, 1;
  4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1;
  5, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  6, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  7, 5, 4, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  8, 6, 5, 4, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;

Jason Kimberley, Table of n, a(n) for n = 1..10126 (rows 1..46)

Cf. A000040, A277648, A278112, A278114, A278115, A278118.

A278112:=func;
A278113_row:=funcA278112(n,p):p in PrimesUpTo(2*n^2)]>;
&cat[A278113_row(n):n in[1..8]];

Table[Floor[n Sqrt[2/Prime@ k]], {n, 8}, {k, PrimePi[2 n^2]}] // Flatten (* Michael De Vlieger, Feb 17 2017 *)

T(n,k) = floor(n*sqrt(2/prime(k))).

T(n,k) sqrt(A000040(k)) <= n sqrt(2) < (T(n,k)+1) sqrt(A000040(k)).

A285786 Number of primes p with 2(n-1)^2 < p <= 2n^2.

Original entry on oeis.org

1, 3, 3, 4, 4, 5, 5, 6, 6, 9, 7, 8, 7, 9, 10, 10, 9, 12, 10, 11, 13, 11, 14, 13, 14, 13, 14, 16, 16, 15, 15, 16, 17, 18, 19, 14, 22, 19, 18, 16, 22, 18, 24, 20, 22, 22, 20, 23, 24, 22, 23, 21, 25, 27, 24, 27, 26, 25, 27, 25, 23, 33, 28, 25, 29, 28, 31, 30, 33, 29

Offset: 1

Ralf Steiner, Apr 26 2017

nonn

The author of the sequence conjectures that a(n) >= 1 for all n. This conjecture is similar to the famous conjecture made by Adrien-Marie Legendre that there is always a prime between n^2 and (n+1)^2, see A014085. - Antti Karttunen, May 01 2017

			For n = 1, the primes from 2*((1-1)^2) to 2*(1^2) (in semiopen range ]0, 2]) are: 2, thus a(1) = 1.
For n = 2, the primes from 2*((2-1)^2) to 2*(2^2) (in semiopen range ]2, 8]) are: 3, 5 and 7, thus a(2) = 3.
For n = 3, the primes from 2*((3-1)^2) to 2*(3^2) (in semiopen range ]8, 18]) are: 11, 13 and 17, thus a(3) = 3.
For n = 4, the primes from 2*((4-1)^2) to 2*(4^2) (in semiopen range ]18, 32]) are: 19, 23, 29 and 31, thus a(4) = 4.

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

Cf. A001105, A000720 (number of primes), A014085 (between n^2 and (n+1)^2), A285738, A285388.

R:= [0, seq(numtheory:-pi(2*n^2),n=1..100)]:
R[2..-1] - R[1..-2]; # Robert Israel, May 01 2017

Table[Length[Select[FactorInteger[Numerator[Table[2^(1 - 2 n^2) n Binomial[2 n^2, n^2], {n, 1, k}]]][[k]][[All, 1]], # > 2 (k - 1)^2 &]], {k, 1, 60}]
Flatten[{1,2,Table[PrimePi[2 k^2] - PrimePi[2 (k - 1)^2], {k, 3, 60}]}]
(* Second program: *)
Array[PrimePi[2 #^2] - PrimePi[2 (# - 1)^2] &, 60] (* Michael De Vlieger, Apr 26 2017, at the suggestion of Robert G. Wilson v. *)

a(n) = (primepi(2*n^2)-primepi(2*(n-1)^2)) \\ David A. Corneth, Apr 27 2017, edited by Antti Karttunen, May 01 2017

a(n)=my(s); forprime(p=2*n^2 - 4*n + 3, 2*n^2, s++); s \\ Charles R Greathouse IV, May 10 2017

from sympy import primepi
def a(n): return primepi(2*n**2) - primepi(2*(n - 1)**2) # Indranil Ghosh, May 01 2017

From Antti Karttunen, May 01 2017: (Start)
a(1) = 1, for n > 1, a(n) = A000720(A001105(n)) - A000720(A001105(n-1)).
For all n except n=2, a(n) <= n.
(End)
First differences of A278114: a(n) = A278114(n) - A278114(n-1) for n > 0, if we use A278114(0) = 0. A278114(n) = Sum_{k=1..n} a(n). - M. F. Hasler, May 02 2017

Definition and value of a(2) changed by Antti Karttunen, May 01 2017

A285269 Number of (odd) primes between 2n^2 and 2(n+1)^2.

Original entry on oeis.org

0, 3, 3, 4, 4, 5, 5, 6, 6, 9, 7, 8, 7, 9, 10, 10, 9, 12, 10, 11, 13, 11, 14, 13, 14, 13, 14, 16, 16, 15, 15, 16, 17, 18, 19, 14, 22, 19, 18, 16, 22, 18, 24, 20, 22, 22, 20, 23, 24, 22, 23, 21, 25, 27, 24, 27, 26, 25, 27, 25, 23, 33, 28, 25, 29, 28, 31, 30, 33, 29

Offset: 0

M. F. Hasler, May 02 2017

nonn

Essentially the same as A285786, except for the offset and initial values.

			a(0) = 0 because between 2*0^2 = 0 and 2*1^2 = 2, there are no (odd) primes.
a(1) = 3 because between 2*1^2 = 2 and 2*2^2 = 8, there are the 3 (odd) primes 3, 5 and 7.
a(2) = 3 because between 2*2^2 = 8 and 2*3^2 = 18, there are the 3 primes 11, 13 and 17.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Join[{0}, Differences[PrimePi[2*Range[100]^2]]] (* Paolo Xausa, Nov 22 2024 *)

a(n)=primepi(2*(n+1)^2-1)-primepi(2*n^2)

a(n) = A285786(n+1), for all n >= 2.

First differences of A278114, a(n) = A278114(n+1) - A278114(n), for n > 0.

cp's OEIS Frontend

A278115 Triangle T(n,k) = A278113(n,k)^2 A000040(k) for 1 <= k <= A278114(n), read by rows.

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A278113 Triangle T(n,k) = A278112(n,A000040(k)) for 1 <= k <= A278114(n), read by rows.

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A285786 Number of primes p with 2(n-1)^2 < p <= 2n^2.

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A285269 Number of (odd) primes between 2*n^2 and 2*(n+1)^2.

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A285269 Number of (odd) primes between 2n^2 and 2(n+1)^2.