1,2

a(p) = p for prime p.

a(n) = n^k when n is the product of k distinct primes (conjecture).

a(n) = n-th term of the n-th Dirichlet self-convolution of the all 1's sequence.

Successive Dirichlet self-convolutions of the all 1's sequence begin:

(1),1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...

1,(2),2,3,2,4,2,4,3,4,2,6,2,4,4,5,... (A000005)

1,3,(3),6,3,9,3,10,6,9,3,18,3,9,9,15,... (A007425)

1,4,4,(10),4,16,4,20,10,16,4,40,4,16,16,35,... (A007426)

1,5,5,15,(5),25,5,35,15,25,5,75,5,25,25,70,... (A061200)

1,6,6,21,6,(36),6,56,21,36,6,126,6,36,36,126,... (A034695)

1,7,7,28,7,49,(7),84,28,49,7,196,7,49,49,210,... (A111217)

1,8,8,36,8,64,8,(120),36,64,8,288,8,64,64,330,... (A111218)

1,9,9,45,9,81,9,165,(45),81,9,405,9,81,81,495,... (A111219)

1,10,10,55,10,100,10,220,55,(100),10,550,10,100,... (A111220)

1,11,11,66,11,121,11,286,66,121,(11),726,11,121,... (A111221)

1,12,12,78,12,144,12,364,78,144,12,(936),12,144,... (A111306)

...

where the main diagonal forms this sequence.

(PARI) {a(n, m=n)=if(n==1, 1, if(m==1, 1, sumdiv(n, d, a(d, 1)*a(n/d, m-1))))}

Sequence in context: A139693 A064946 A078730 this_sequence A128531 A123167 A141670

Adjacent sequences: A163764 A163765 A163766 this_sequence A163768 A163769 A163770

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Aug 04 2009