A284061 Triangle read by rows: T(n,k) = pi(prime(k) * prime(n+1)).

3, 4, 6, 6, 8, 11, 8, 11, 16, 21, 9, 12, 18, 24, 34, 11, 15, 23, 30, 42, 47, 12, 16, 24, 32, 46, 53, 66, 14, 19, 30, 37, 54, 62, 77, 84, 16, 23, 34, 46, 66, 74, 94, 101, 121, 18, 24, 36, 47, 68, 79, 99, 107, 127, 154, 21, 29, 42, 55, 79, 92, 114, 126, 146, 180

Offset

1,1

Comments

Prime(T(n,k)) is the largest prime q such that q * p_n# / prime(k) < p_(n+1)#, with primorial p_n# = A002110(n).

T(n,1) = A020900(n+1), T(n,2) = A020901(n+1), T(n,3) = A020935(n+1), T(n,4) = A020937(n+1).

Links

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150).

Example

a(5) = T(2,2) = 8 since the largest prime q <= prime(2) prime(3+1) = 3*7 = 21 is 19, the 8th prime.
Rows 1 <= n <= 12 of triangle T(n,k):
   3
   4    6
   6    8   11
   8   11   16   21
   9   12   18   24   34
  11   15   23   30   42   47
  12   16   24   32   46   53    66
  14   19   30   37   54   62    77    84
  16   23   34   46   66   74    94   101   121
  18   24   36   47   68   79    99   107   127   154
  21   29   42   55   79   92   114   126   146   180   189
  22   30   46   61   87   99   125   137   160   195   205   240
Values of m = q * p_n#/prime(k) < p_(n+1)# with q = prime(T(n,k)):
                                    prime(k)
                      2       3       5       7      11      13
           6  |       5
          30  |      21      26
p_(n+1)# 210  |     195     190     186
        2310  |    1995    2170    2226    2190
       30030  |   26565   28490   28182   29370   29190
      510510  |  465465  470470  498498  484770  494130  487410
All terms m of row n have omega(m) = A001221(m) = n.

Mathematica

Table[PrimePi[Prime[k] Prime[n + 1]], {n, 11}, {k, n}] // Flatten

Prog

(PARI) for(n=1, 12, for(k=1, n, print1(primepi(prime(k) * prime(n + 1)), ", "); ); print(); ); \\ Indranil Ghosh, Mar 19 2017
(Python)
from sympy import prime, primepi
for n in range(1, 13):
    print([primepi(prime(k) * prime(n + 1)) for k in range(1, n+1)])
# Indranil Ghosh, Mar 19 2017

Crossrefs

Cf. A020900, A020901, A020935, A020937.

Sequence in context: A347318 A275883 A274529 * A162625 A033095 A158907

Adjacent sequences: A284058 A284059 A284060 * A284062 A284063 A284064

Keyword

nonn,tabl,easy

Author

Michael De Vlieger, Mar 19 2017

Status

approved