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A373845 Triangle read by rows: T(n,k) = arithmetic derivative of (1 + A002110(n) + A002110(k)), 1 <= k <= n, where A002110(n) is the n-th primorial number.

%I A373845 #15 Jun 21 2024 14:17:46
%S A373845 1,6,1,14,1,1,74,38,1,1,1551,338,1,1,1,21084,8631,1330,1,1,3550,
%T A373845 172655,72938,1970,3410,1,1,5822,3233234,4157356,421750,228491,10190,
%U A373845 13610,537398,289610,297753138,32805527,5188250,8698439,761710,1,18344100,1,6954431,2156564414,929540471,68769335,335525472,4283242,21900155,348965439,109820278,185002,32593310
%N A373845 Triangle read by rows: T(n,k) = arithmetic derivative of (1 + A002110(n) + A002110(k)), 1 <= k <= n, where A002110(n) is the n-th primorial number.
%C A373845 Arithmetic derivatives of the sums of three primorials, of which one is 1 [= A002110(0)], and two are > 1.
%C A373845 Ones occur in positions where 1 + A002110(n) + A002110(k) is a prime.
%C A373845 See also comments in A373844, and in A373848.
%H A373845 Antti Karttunen, <a href="/A373845/b373845.txt">Table of n, a(n) for n = 1..1035; the first 45 rows of the triangle</a>
%H A373845 <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%F A373845 For n, k >= 1, T(n, k) = A003415(1+A370121(n, k)).
%e A373845 Triangle begins as:
%e A373845         1,
%e A373845         6,        1,
%e A373845        14,        1,       1,
%e A373845        74,       38,       1,       1,
%e A373845      1551,      338,       1,       1,      1,
%e A373845     21084,     8631,    1330,       1,      1,  3550,
%e A373845    172655,    72938,    1970,    3410,      1,     1,     5822,
%e A373845   3233234,  4157356,  421750,  228491,  10190, 13610,   537398, 289610,
%e A373845 297753138, 32805527, 5188250, 8698439, 761710,     1, 18344100,      1, 6954431,
%e A373845 etc.
%o A373845 (PARI)
%o A373845 A002110(n) = prod(i=1,n,prime(i));
%o A373845 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A373845 A373845(n) = { n--; my(c = (sqrtint(8*n + 1) - 1) \ 2, x=A002110(1+n - binomial(c + 1, 2))); A003415(1+(A002110(1+c)+x)); };
%Y A373845 Cf. A002110, A003415, A370121, A373844, A373848.
%Y A373845 Cf. also A024451, A370129, A370138 (arithmetic derivative applied to the sums of a constant number of primorials).
%K A373845 nonn,tabl
%O A373845 1,2
%A A373845 _Antti Karttunen_, Jun 21 2024