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

%I #22 Mar 19 2023 23:37:20

%S 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,

%T 46,53,66,14,19,30,37,54,62,77,84,16,23,34,46,66,74,94,101,121,18,24,

%U 36,47,68,79,99,107,127,154,21,29,42,55,79,92,114,126,146,180

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

%C 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).

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

%H Michael De Vlieger, <a href="/A284061/b284061.txt">Table of n, a(n) for n = 1..11325</a> (rows 1 <= n <= 150).

%e a(5) = T(2,2) = 8 since the largest prime q <= prime(2) prime(3+1) = 3*7 = 21 is 19, the 8th prime.

%e Rows 1 <= n <= 12 of triangle T(n,k):

%e 3

%e 4 6

%e 6 8 11

%e 8 11 16 21

%e 9 12 18 24 34

%e 11 15 23 30 42 47

%e 12 16 24 32 46 53 66

%e 14 19 30 37 54 62 77 84

%e 16 23 34 46 66 74 94 101 121

%e 18 24 36 47 68 79 99 107 127 154

%e 21 29 42 55 79 92 114 126 146 180 189

%e 22 30 46 61 87 99 125 137 160 195 205 240

%e Values of m = q * p_n#/prime(k) < p_(n+1)# with q = prime(T(n,k)):

%e prime(k)

%e 2 3 5 7 11 13

%e 6 | 5

%e 30 | 21 26

%e p_(n+1)# 210 | 195 190 186

%e 2310 | 1995 2170 2226 2190

%e 30030 | 26565 28490 28182 29370 29190

%e 510510 | 465465 470470 498498 484770 494130 487410

%e All terms m of row n have omega(m) = A001221(m) = n.

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

%o (PARI) for(n=1, 12, for(k=1, n, print1(primepi(prime(k) * prime(n + 1)),", ");); print();); \\ _Indranil Ghosh_, Mar 19 2017

%o (Python)

%o from sympy import prime, primepi

%o for n in range(1, 13):

%o print([primepi(prime(k) * prime(n + 1)) for k in range(1, n+1)])

%o # _Indranil Ghosh_, Mar 19 2017

%Y Cf. A020900, A020901, A020935, A020937.

%K nonn,tabl,easy

%O 1,1

%A _Michael De Vlieger_, Mar 19 2017