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
Examples
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.
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150).
Programs
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Mathematica
Table[PrimePi[Prime[k] Prime[n + 1]], {n, 11}, {k, n}] // Flatten -
PARI
for(n=1, 12, for(k=1, n, print1(primepi(prime(k) * prime(n + 1)),", ");); print();); \\ Indranil Ghosh, Mar 19 2017
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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
Comments