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A138618 Triangle of exponentials of Mangoldt function M(n) read by rows, in which row products give the natural numbers.

%I A138618 #25 Feb 16 2025 08:33:08
%S A138618 1,2,1,3,1,1,2,2,1,1,5,1,1,1,1,1,3,2,1,1,1,7,1,1,1,1,1,1,2,2,1,2,1,1,
%T A138618 1,1,3,1,3,1,1,1,1,1,1,1,5,1,1,2,1,1,1,1,1,11,1,1,1,1,1,1,1,1,1,1,1,1,
%U A138618 2,3,1,2,1,1,1,1,1,1,13,1,1,1,1,1,1,1,1,1,1,1,1,1,7,1,1,1,1,2,1,1,1,1,1,1,1
%N A138618 Triangle of exponentials of Mangoldt function M(n) read by rows, in which row products give the natural numbers.
%C A138618 Row sums are A001414. This table is similar to A139547 and A120885.
%C A138618 Cumulative column products are A003418, A139550, A139552, A139554.
%H A138618 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MangoldtFunction.html">Mangoldt Function</a>.
%F A138618 T(n,k) = A014963(n/k) if n mod k = 0, otherwise 1. - _Mats Granvik_, May 23 2013
%e A138618 1 = 1
%e A138618 2*1 = 2
%e A138618 3*1*1 = 3
%e A138618 2*2*1*1 = 4
%e A138618 5*1*1*1*1 = 5
%e A138618 1*3*2*1*1*1 = 6
%e A138618 7*1*1*1*1*1*1 = 7
%e A138618 2*2*1*2*1*1*1*1 = 8
%e A138618 3*1*3*1*1*1*1*1*1 = 9
%e A138618 1*5*1*1*2*1*1*1*1*1 = 10
%e A138618 11*1*1*1*1*1*1*1*1*1*1 = 11
%e A138618 1*1*2*3*1*2*1*1*1*1*1*1 = 12
%e A138618 13*1*1*1*1*1*1*1*1*1*1*1*1 = 13
%t A138618 Flatten[Table[Table[If[Mod[n, k] == 0, Exp[MangoldtLambda[n/k]], 1], {k, 1, n}], {n, 1, 14}]] (* _Mats Granvik_, May 23 2013 *)
%o A138618 (PARI) M(n) = ispower(n, , &n); if(isprime(n), n, 1); \\ A014963
%o A138618 T(n,k) = if (n % k, 1, M(n/k));
%o A138618 row(n) = vector(n, k, T(n,k)); \\ _Michel Marcus_, Mar 03 2023
%Y A138618 Cf. A120885, A139547, A001414, A000027.
%K A138618 nonn,tabl
%O A138618 1,2
%A A138618 _Mats Granvik_, May 14 2008