A278118 -id:A278118 - cp's OEIS frontend

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

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)).

A278116 a(n) is the largest j such that A278115(n,k) strictly decreases for k=1..j.

Original entry on oeis.org

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

Offset: 1

Jason Kimberley, Feb 12 2017

Jason Kimberley, Table of n, a(n) for n = 1..100000

Cf. A278102.
This is the row length sequence for triangles A278117 and A278118.
A278119 lists first occurrences in this sequence.

A:=func;
A278116:=funcA278115(n,P[j+1])}
  select j else #P) where P is PrimesUpTo(2*n^2)>;
[A278116(n):n in[1..103]];

Map[1 + Length@ TakeWhile[Differences@ #, # < 0 &] &, #] &@ Table[# Floor[n Sqrt[2/#]]^2 &@ Prime@ k, {n, 105}, {k, PrimePi[2 n^2]}] (* Michael De Vlieger, Feb 17 2017 *)

def isqrt(n):
    if n < 0:
        raise ValueError('imaginary')
    if n == 0:
        return 0
    a, b = divmod(n.bit_length(),2)
    x = 2**(a+b)
    while True:
        y = (x + n//x)//2
        if y >= x:
            return x
        x = y;
def next_prime(n):
    for p in range(n+1,2*n+1):
        for i in range(2,isqrt(n)+1):
            if p % i == 0:
                break
        else:
            return p
    return None
def A278116(n):
    k = 0
    p = 2
    s2= (n**2)*p
    s = s2
    while True:
        s_= s
        k+= 1
        p = next_prime(p)
        s = (isqrt(s2//p)**2)*p
        if s > s_:
            break
    return k

A278117 Irregular triangle T(n,k) = A278115(n,k) for 1 <= k <= A278116(n), read by rows.

Original entry on oeis.org

2, 8, 3, 18, 12, 5, 32, 27, 20, 50, 48, 45, 28, 72, 48, 45, 98, 75, 128, 108, 162, 147, 125, 112, 99, 200, 192, 180, 175, 242, 192, 180, 175, 288, 243, 338, 300, 392, 363, 320, 450, 432, 405, 512, 507, 500, 448, 396, 578, 507, 500, 648, 588, 722, 675, 800, 768, 720

Offset: 1

Jason Kimberley, Feb 12 2017

Each row is the longest strictly decreasing prefix of the corresponding row of A278115.

			The first six rows are:
2;
8, 3;
18, 12, 5;
32, 27, 20;
50, 48, 45, 28;
72, 48, 45;

Jason Kimberley, Table of i, a(i) for i = 1..10104 (T(n,k) for n = 1..3333)

Cf. A278103, A278118.

A278112:=func;
A278115_row:=funcA278112(n,p)^2*p:p in PrimesUpTo(2*n^2)]>;
A278117_row:=funcA278115_row(n) >;
&cat[A278117_row(n):n in[1..20]];

Map[Take[#, 1 + Length@ TakeWhile[Differences@ #, # < 0 &]] &, #] &@ Table[# Floor[n Sqrt[2/#]]^2 &@ Prime@ k, {n, 20}, {k, PrimePi[2 n^2]}] // Flatten (* Michael De Vlieger, Feb 17 2017 *)

cp's OEIS Frontend

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

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A278116 a(n) is the largest j such that A278115(n,k) strictly decreases for k=1..j.

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A278117 Irregular triangle T(n,k) = A278115(n,k) for 1 <= k <= A278116(n), read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica