A377992 - OEIS

A377992

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

%I #24 Nov 22 2024 20:32:14

%S 6,30,58,210,435,507,2310,8435,21827,29233,30030,39030,62762,69914,

%T 76442,78874,510510,1342785,1958673,9699690,28235362,223092870,

%U 975351895,1527890095,1885679383,2189118743,2329696457,2338611863,3485765789,4586671213,5453593183,5472849253,5674340053,8071055747,8931775397,9332889127

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

%C 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.

%C 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).

%e The initial rows of the triangle:

%e Row n terms

%e 2 6;

%e 3 30, 58;

%e 4 210, 435, 507;

%e 5 2310, 8435, 21827, 29233;

%e 6 30030, 39030, 62762, 69914, 76442, 78874;

%e 7 510510, 1342785, 1958673;

%e 8 9699690, 28235362;

%e 9 223092870, 975351895, 1527890095, ..., , 1167539981207, 1171314743479; (row 9 has 330 terms that are given separately in A378209)

%e 10 6469693230, 27623935255, 37262208055;

%e 11 200560490130, 345634019382, 440192669882;

%e etc.

%e 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.

%e Note that 507 is so far the only known term in this triangle that is not squarefree (in A005117).

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

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

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

%Y Cf. also A327978, A366890, A369240, A377987.

%K nonn,tabf

%O 2,1

%A _Antti Karttunen_, Nov 19 2024