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.

1, 1, 4, 1, 12, 16, 1, 80, 60, 92, 1, 216, 540, 608, 704, 1, 3740, 3100, 4548, 6324, 8164, 568, 60080, 40060, 56292, 116208, 61768, 110752, 33975, 1021040, 1041768, 794468, 2415104, 1091004, 1357128, 1942844, 28300, 9789116, 29099520, 19722884, 18576860, 35347200, 35779644, 26575580, 37935056, 704080, 335024060

Offset

0,3

Comments

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.

Links

Antti Karttunen, Table of n, a(n) for n = 0..1377; the first 52 rows of triangle, flattened

Index entries for sequences related to primorial numbers

Formula

a(n) = A003415(A370121(n)).

For n, k >= 1, T(n,k) = A002110(k)*A370136(n,k) + A024451(k)*A370135(n,k).

Example

Triangle begins as:
      1;
      1,       4;
      1,      12,       16;
      1,      80,       60,       92;
      1,     216,      540,      608,      704;
      1,    3740,     3100,     4548,     6324,     8164;
    568,   60080,    40060,    56292,   116208,    61768,   110752;
  33975, 1021040,  1041768,   794468,  2415104,  1091004,  1357128,  1942844;
  28300, 9789116, 29099520, 19722884, 18576860, 35347200, 35779644, 26575580, 37935056;

Prog

(PARI)
A002110(n) = prod(i=1, n, prime(i));
A370121(n) = { my(c = (sqrtint(8*n + 1) - 1) \ 2); (A002110(c) + A002110(n - binomial(c + 1, 2))); };
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A370129(n) = A003415(A370121(n));

Crossrefs

Cf. A002110, A003415, A024451, A370121, A370135, A370136.

Cf. A014545 (positions of 1's at the left edge), A087112.

Cf. also A024451 (arithmetic derivatives of primorials).

Sequence in context: A338864 A078219 A373547 * A187541 A117413 A322970

Adjacent sequences: A370126 A370127 A370128 * A370130 A370131 A370132

Keyword

nonn,tabl

Author

Antti Karttunen, Feb 29 2024

Status

approved