A377992 Irregular triangle giving on row n all antiderivatives of A024451(n), for n >= 2.

6, 30, 58, 210, 435, 507, 2310, 8435, 21827, 29233, 30030, 39030, 62762, 69914, 76442, 78874, 510510, 1342785, 1958673, 9699690, 28235362, 223092870, 975351895, 1527890095, 1885679383, 2189118743, 2329696457, 2338611863, 3485765789, 4586671213, 5453593183, 5472849253, 5674340053, 8071055747, 8931775397, 9332889127

Offset: 2

Author

Antti Karttunen, Nov 19 2024

Keywords

• nonn
• tabf

Comments

Row n lists in ascending order all numbers k whose arithmetic derivative k' [A003415(k)] is equal to A024451(n) = A003415(A002110(n)). For A024451(1) = 1, there is an infinite number of integers k for which A003415(k) = 1 (namely, all the primes), therefore the rows start from index n=2, with each having A377993(n) terms. Note that as a whole, this sequence is not monotonic, for example, the last term on row 9, 1171314743479 is larger than the first term of row 10, 6469693230.

Because A024451 is a subsequence of A048103, this is also. And if all terms of A024451 are squarefree as is conjectured, then all terms of this sequence are cubefree (A004709).

Examples

			The initial rows of the triangle:
Row n  terms
   2   6;
   3   30, 58;
   4   210, 435, 507;
   5   2310, 8435, 21827, 29233;
   6   30030, 39030, 62762, 69914, 76442, 78874;
   7   510510, 1342785, 1958673;
   8   9699690, 28235362;
   9   223092870, 975351895, 1527890095, ..., , 1167539981207, 1171314743479; (row 9 has 330 terms that are given separately in A378209)
  10   6469693230, 27623935255, 37262208055;
  11   200560490130, 345634019382, 440192669882;
  etc.
The only terms that occur on row 4 are k = 210, 435, 507 ( = 2*3*5*7, 3*5*29, 3 * 13^2) as they are only numbers for which A003415(k) = 247 = A024451(4) = A003415(A002110(4)), as we have (2*3*5*7)' = (3*5)'*(2*7) + (2*7)'*3*5 = (8*14) + (9*15) = (3*5*29)' = (3*5)'*29 + (3*5)*29' = (8*29 + 15*1) = (3 * 13 * 13)' = (3*13)'*13 + (3*13)*13' = 16*13 + 3*13*1 = 19*13 = 247.
 Note that 507 is so far the only known term in this triangle that is not squarefree (in A005117).
		

Crossrefs

Cf. A003415, A005117, A024451, A377993 (row lengths).

Subsequence of A048103, conjectured also to be a subsequence of A004709.

Cf. A002110 (left edge, from its term a(2)=6 onward), A378209 (row 9).

Cf. also A327978, A366890, A369240, A377987.