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

1,2

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.

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

Formula

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

Example

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.

Prog

(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)); };

Crossrefs

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

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

Sequence in context: A122508 A171006 A176121 * A062190 A080211 A146997

Adjacent sequences: A373842 A373843 A373844 * A373846 A373847 A373848

Keyword

nonn,tabl

Author

Antti Karttunen, Jun 21 2024

Status

approved