A370136 - OEIS

%I #7 Mar 08 2024 16:11:47

%S 1,4,1,32,5,1,55,60,12,1,1292,195,71,16,1,22532,2505,841,384,9,1,

%T 382892,102723,8897,8640,191,21,1,2469635,3502740,323328,34133,9980,

%U 756,24,1,111738812,18755325,10308201,1568312,50621,5211,371,44,1,4853127108,2003156919,107924801,178347008,2376149,251367,6339,672,31,1

%N Triangle read by rows: T(n,k) = arithmetic derivative of ((A002110(n) + A002110(k)) / A002110(k)), 1 <= k <= n.

%H Antti Karttunen, <a href="/A370136/b370136.txt">Table of n, a(n) for n = 1..1081; the first 46 rows of triangle, flattened</a>

%H <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>

%F a(n) = A003415(A370135(n)).

%e Triangle begins as:

%e           1;

%e           4,        1;

%e          32,        5,        1;

%e          55,       60,       12,       1;

%e        1292,      195,       71,      16,     1;

%e       22532,     2505,      841,     384,     9,    1;

%e      382892,   102723,     8897,    8640,   191,   21,   1;

%e     2469635,  3502740,   323328,   34133,  9980,  756,  24,  1;

%e   111738812, 18755325, 10308201, 1568312, 50621, 5211, 371, 44, 1;

%o (PARI)

%o A002110(n) = prod(i=1,n,prime(i));

%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

%o A370135(n) = { n--; my(c = (sqrtint(8*n + 1) - 1) \ 2, x=A002110(1+n - binomial(c + 1, 2))); ((A002110(1+c)+x)/x); };

%o A370136(n) = A003415(A370135(n));

%Y Cf. A002110, A003415, A370129, A370135.

%K nonn,tabl

%O 1,2

%A _Antti Karttunen_, Mar 07 2024