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

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, 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, 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

Offset: 1

Author

Mats Granvik, May 14 2008

Keywords

• nonn
• tabl

Comments

Row sums are A001414. This table is similar to A139547 and A120885.

Cumulative column products are A003418, A139550, A139552, A139554.

Examples

			1 = 1
2*1 = 2
3*1*1 = 3
2*2*1*1 = 4
5*1*1*1*1 = 5
1*3*2*1*1*1 = 6
7*1*1*1*1*1*1 = 7
2*2*1*2*1*1*1*1 = 8
3*1*3*1*1*1*1*1*1 = 9
1*5*1*1*2*1*1*1*1*1 = 10
11*1*1*1*1*1*1*1*1*1*1 = 11
1*1*2*3*1*2*1*1*1*1*1*1 = 12
13*1*1*1*1*1*1*1*1*1*1*1*1 = 13
		

Links

• Eric Weisstein's World of Mathematics, Mangoldt Function.

Crossrefs

Cf. A120885, A139547, A001414, A000027.

Programs

Mathematica

Flatten[Table[Table[If[Mod[n, k] == 0, Exp[MangoldtLambda[n/k]], 1], {k, 1, n}], {n, 1, 14}]] (* Mats Granvik, May 23 2013 *)

PARI

M(n) = ispower(n, , &n); if(isprime(n), n, 1); \\ A014963
T(n,k) = if (n % k, 1, M(n/k));
row(n) = vector(n, k, T(n,k)); \\ Michel Marcus, Mar 03 2023

Formula

T(n,k) = A014963(n/k) if n mod k = 0, otherwise 1. - Mats Granvik, May 23 2013