A378133 - OEIS

A378133

Irregular triangle T(n,k) = P(n)*2^k, n >= 0, k = 0..floor(log_2 prime(k+1)), where P = A002110.

%I #5 Nov 19 2024 01:03:19

%S 1,2,4,6,12,24,30,60,120,210,420,840,1680,2310,4620,9240,18480,30030,

%T 60060,120120,240240,480480,510510,1021020,2042040,4084080,8168160,

%U 9699690,19399380,38798760,77597520,155195040,223092870,446185740,892371480,1784742960,3569485920

%N Irregular triangle T(n,k) = P(n)*2^k, n >= 0, k = 0..floor(log_2 prime(k+1)), where P = A002110.

%C Subset of A060735.

%H Michael De Vlieger, <a href="/A378133/b378133.txt">Table of n, a(n) for n = 0..3494</a> (* rows n = 0..349, flattened *)

%F T(n,k) = A002110(n)*A000079(k), n >= 0, k = 0..A098388(k+1).

%F T(n,0) = A002110(n).

%F T(n,1) = A088860(n), n >= 1.

%F T(n,2) = A102476(n), n >= 2.

%F T(n,A098388(k+1)) = A378144(n).

%F Let S(n,j) = A002110(n)*j, n >= 0, j = 0..A006093(n+1) = P(n)*j, n >= 0, j = 0..prime(n+1)-1. Then T(n,k) = S(n, 2^k).

%e Rows n = 0..9:

%e n\k | 0 1 2 3 4

%e -------------------------------------------------------------

%e 0 | 1 . . . .

%e 1 | 2 4 . . .

%e 2 | 6 12 24 . .

%e 3 | 30 60 120 . .

%e 4 | 210 420 840 1680 .

%e 5 | 2310 4620 9240 18480 .

%e 6 | 30030 60060 120120 240240 480480

%e 7 | 510510 1021020 2042040 4084080 8168160

%e 8 | 9699690 19399380 38798760 77597520 155195040

%e 9 | 223092870 446185740 892371480 1784742960 3569485920

%t nn = 16;

%t MapIndexed[Set[P[First[#2] - 1], #1] &,

%t FoldList[Times, 1, Prime@ Range[nn + 1] ] ];

%t Union@ Flatten@

%t Table[P[i]*2^Range[0, Floor[Log2[Prime[i + 1] ] ] ], {i, 0, nn}]

%Y Cf. A000079, A002110, A060735, A088860, A098388, A102476, A378144.

%K nonn,tabf,easy

%O 0,2

%A _Michael De Vlieger_, Nov 17 2024