1,6

Conjectures: A) a(n) is always an integer. B) If p is a prime then p|a(n) if and only if p <= n/3. Let ordp(n,p) denote the exponent of the largest power of p which divides n. For example, ordp(48,2)=4 since 48=3*(2^4). The precise decomposition of a(n) into primes would follow from the following two conjectures: C) For each positive integer n and prime p, ordp(a(np),p)= ordp(a(np+1),p)= ordp(a(np+2),p)= . . . = ordp(a(np+p-1),p). D) Let b(n)=A004125(n). Then ordp(a(np),p)= b(n)+ b(floor(n/p))+ b(floor(n/p^2))+ b(floor(n/p^3))+ . . .. This is reminiscent of de Polignac's formula (also due to Legendre) for the prime factorization of n! (see the link).

De Polignac's formula, Link to Wikipedia entry.

a(n)=(product{j=1..n}product{k=1..n} gcd(j,k))/(product{j=1..n}product{d|j} d^(j/d)). a(n)=(product{j=1..n}product{k=1..n}gcd(j,k))/(product{k=1..n}(floor(n/k)!)^k).

Cf. A004125, A092287, A129364.

Sequence in context: A025248 A101416 A098920 this_sequence A021453 A053789 A115596

Adjacent sequences: A129362 A129363 A129364 this_sequence A129366 A129367 A129368

easy,nonn

Peter Bala (pbala(AT)toucansurf.com), Apr 13 2007