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%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
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