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A370129 Triangle read by rows: T(n,k) = A003415(A002110(n)+A002110(k)), 0 <= k <= n; arithmetic derivatives of the sums of two primorial numbers.

%I A370129 #17 Mar 08 2024 16:11:09
%S A370129 1,1,4,1,12,16,1,80,60,92,1,216,540,608,704,1,3740,3100,4548,6324,
%T A370129 8164,568,60080,40060,56292,116208,61768,110752,33975,1021040,1041768,
%U A370129 794468,2415104,1091004,1357128,1942844,28300,9789116,29099520,19722884,18576860,35347200,35779644,26575580,37935056,704080,335024060
%N A370129 Triangle read by rows: T(n,k) = A003415(A002110(n)+A002110(k)), 0 <= k <= n; arithmetic derivatives of the sums of two primorial numbers.
%C A370129 Apart from those positions (A014545) at the left edge where a(n) = 1, a(n) <= A087112(1+n) only at n=2, 4 and 5, i.e., never after the third row.
%H A370129 Antti Karttunen, <a href="/A370129/b370129.txt">Table of n, a(n) for n = 0..1377; the first 52 rows of triangle, flattened</a>
%H A370129 <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%F A370129 a(n) = A003415(A370121(n)).
%F A370129 For n, k >= 1, T(n,k) = A002110(k)*A370136(n,k) + A024451(k)*A370135(n,k).
%e A370129 Triangle begins as:
%e A370129       1;
%e A370129       1,       4;
%e A370129       1,      12,       16;
%e A370129       1,      80,       60,       92;
%e A370129       1,     216,      540,      608,      704;
%e A370129       1,    3740,     3100,     4548,     6324,     8164;
%e A370129     568,   60080,    40060,    56292,   116208,    61768,   110752;
%e A370129   33975, 1021040,  1041768,   794468,  2415104,  1091004,  1357128,  1942844;
%e A370129   28300, 9789116, 29099520, 19722884, 18576860, 35347200, 35779644, 26575580, 37935056;
%o A370129 (PARI)
%o A370129 A002110(n) = prod(i=1,n,prime(i));
%o A370129 A370121(n) = { my(c = (sqrtint(8*n + 1) - 1) \ 2); (A002110(c) + A002110(n - binomial(c + 1, 2))); };
%o A370129 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A370129 A370129(n) = A003415(A370121(n));
%Y A370129 Cf. A002110, A003415, A024451, A370121, A370135, A370136.
%Y A370129 Cf. A014545 (positions of 1's at the left edge), A087112.
%Y A370129 Cf. also A024451 (arithmetic derivatives of primorials).
%K A370129 nonn,tabl
%O A370129 0,3
%A A370129 _Antti Karttunen_, Feb 29 2024