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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.
3

%I #15 Jun 21 2024 14:17:46

%S 1,6,1,14,1,1,74,38,1,1,1551,338,1,1,1,21084,8631,1330,1,1,3550,

%T 172655,72938,1970,3410,1,1,5822,3233234,4157356,421750,228491,10190,

%U 13610,537398,289610,297753138,32805527,5188250,8698439,761710,1,18344100,1,6954431,2156564414,929540471,68769335,335525472,4283242,21900155,348965439,109820278,185002,32593310

%N 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 Arithmetic derivatives of the sums of three primorials, of which one is 1 [= A002110(0)], and two are > 1.

%C Ones occur in positions where 1 + A002110(n) + A002110(k) is a prime.

%C See also comments in A373844, and in A373848.

%H 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 <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>

%F For n, k >= 1, T(n, k) = A003415(1+A370121(n, k)).

%e Triangle begins as:

%e 1,

%e 6, 1,

%e 14, 1, 1,

%e 74, 38, 1, 1,

%e 1551, 338, 1, 1, 1,

%e 21084, 8631, 1330, 1, 1, 3550,

%e 172655, 72938, 1970, 3410, 1, 1, 5822,

%e 3233234, 4157356, 421750, 228491, 10190, 13610, 537398, 289610,

%e 297753138, 32805527, 5188250, 8698439, 761710, 1, 18344100, 1, 6954431,

%e etc.

%o (PARI)

%o A002110(n) = prod(i=1,n,prime(i));

%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

%o 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 Cf. A002110, A003415, A370121, A373844, A373848.

%Y Cf. also A024451, A370129, A370138 (arithmetic derivative applied to the sums of a constant number of primorials).

%K nonn,tabl

%O 1,2

%A _Antti Karttunen_, Jun 21 2024