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.

1, 6, 1, 14, 1, 1, 74, 38, 1, 1, 1551, 338, 1, 1, 1, 21084, 8631, 1330, 1, 1, 3550, 172655, 72938, 1970, 3410, 1, 1, 5822, 3233234, 4157356, 421750, 228491, 10190, 13610, 537398, 289610, 297753138, 32805527, 5188250, 8698439, 761710, 1, 18344100, 1, 6954431, 2156564414, 929540471, 68769335, 335525472, 4283242, 21900155, 348965439, 109820278, 185002, 32593310

Offset: 1

Author

Antti Karttunen, Jun 21 2024

Keywords

• nonn
• tabl

Comments

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

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

See also comments in A373844, and in A373848.

Examples

			Triangle begins as:
        1,
        6,        1,
       14,        1,       1,
       74,       38,       1,       1,
     1551,      338,       1,       1,      1,
    21084,     8631,    1330,       1,      1,  3550,
   172655,    72938,    1970,    3410,      1,     1,     5822,
  3233234,  4157356,  421750,  228491,  10190, 13610,   537398, 289610,
297753138, 32805527, 5188250, 8698439, 761710,     1, 18344100,      1, 6954431,
etc.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..1035; the first 45 rows of the triangle
• Index entries for sequences related to primorial numbers

Crossrefs

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

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

Programs

PARI

A002110(n) = prod(i=1,n,prime(i));
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
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)); };

Formula

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