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A253635 Rectangular array read by upwards antidiagonals: a(n,k) = index of largest term <= 10^k in row n of A253572, n >= 1, k >= 0.

%I A253635 #22 Dec 30 2019 19:39:01
%S A253635 1,1,4,1,7,7,1,9,20,10,1,10,34,40,14,1,10,46,86,67,17,1,10,55,141,175,
%T A253635 101,20,1,10,62,192,338,313,142,24,1,10,67,242,522,694,507,190,27,1,
%U A253635 10,72,287,733,1197,1273,768,244,30
%N A253635 Rectangular array read by upwards antidiagonals: a(n,k) = index of largest term <= 10^k in row n of A253572, n >= 1, k >= 0.
%C A253635 Or a(n,k) = the number of positive integers less than or equal to 10^k that are divisible by no prime exceeding prime(n).
%e A253635 Array begins:
%e A253635 {1,  4,  7,  10,   14,   17,    20,    24,    27,     30, ...}
%e A253635 {1,  7, 20,  40,   67,  101,   142,   190,   244,    306, ...}
%e A253635 {1,  9, 34,  86,  175,  313,   507,   768,  1105,   1530, ...}
%e A253635 {1, 10, 46, 141,  338,  694,  1273,  2155,  3427,   5194, ...}
%e A253635 {1, 10, 55, 192,  522, 1197,  2432,  4520,  7838,  12867, ...}
%e A253635 {1, 10, 62, 242,  733, 1848,  4106,  8289, 15519,  27365, ...}
%e A253635 {1, 10, 67, 287,  945, 2579,  6179, 13389, 26809,  50351, ...}
%e A253635 {1, 10, 72, 331, 1169, 3419,  8751, 20198, 42950,  85411, ...}
%e A253635 {1, 10, 76, 369, 1385, 4298, 11654, 28434, 63768, 133440, ...}
%e A253635 {1, 10, 79, 402, 1581, 5158, 14697, 37627, 88415, 193571, ...}
%t A253635 r = 10; y[1] = t = Table[2^j, {j, 0, 39}]; max = 10^13; len = 10^10; prev = 0; For[n = 2, n <= r, n++, next = 0; For[k = 1, k <= 43, k++, If[Prime[n]^k < max, t = Union[t, Prime[n]*t]; s = FirstPosition[t, v_ /; v > len, 0]; t = Take[t, s[[1]] - 1]; If[t[[-1]] > len, t = Delete[t, -1]]; next = Length[t]; If[next == prev, Break, prev = next], Break]]; y[n] = t]; b[i_, j_] := FirstPosition[y[i], v_ /; v > 10^j][[1]]; a253635[n_, j_] := If[IntegerQ[b[n, j]], b[n, j] - 1, 0]; Flatten[Table[a253635[n - j, j], {n, r}, {j, 0, n - 1}]] (* array antidiagonals flattened *)
%Y A253635 Cf. A123384, A100752, A106598, A106600, A107352, A106629, A108275, A108276, A108277 (rows 1-9).
%Y A253635 Cf. A011557, A066343, A253572, A253573.
%K A253635 nonn,tabl
%O A253635 1,3
%A A253635 _L. Edson Jeffery_, Jan 07 2015