A368336 -id:A368336 - cp's OEIS frontend

A368331 The number of divisors of the largest term of A054743 that divides of n.

1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 21 2023

First differs from A366145 at n = 27.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A054743, A207481, A368328, A368329, A368330, A368335, A368336.

Cf. A366145.

f[p_, e_] := If[e <= p, 1, e+1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] <= f[i,1], 1, f[i,2]+1));}

Multiplicative with a(p^e) = 1 if e <= p, and a(p^e) = e+1 if e > p.
a(n) = A000005(A368329(n)).
a(n) >= 1, with equality if and only if n is in A207481.
a(n) <= A000005(n), with equality if and only if n is in A054743.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s - 1/p^((p+2)*s-1) + 1/p^((p+1)*s) + 1/p^((p+1)*s-1)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/((p-1)*p^(p-1))) = 1.58396891058853238595... .

A368335 The number of divisors of the largest term of A054744 that divides of n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 4, 3, 1, 1, 1, 6, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 4, 1, 4, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 5, 5, 1, 1, 3, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 21 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A048103, A054744, A368331, A368332, A368333, A368334, A368336.

f[p_, e_] := If[e < p, 1, e+1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] < f[i,1], 1, f[i,2]+1));}

Multiplicative with a(p^e) = 1 if e < p, and a(p^e) = e+1 if e >= p.
a(n) = A000005(A368333(n)).
a(n) >= 1, with equality if and only if n is in A048103.
a(n) <= A000005(n), with equality if and only if n is in A054744.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + 1/p^(p*s-1) + 1/p^((p+1)*s) - 1/p^((p+1)*s-1)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + (1 + (p-1)*p)/((p-1)*p^p)) = 1.98019019497523582894... .

cp's OEIS Frontend

A368331 The number of divisors of the largest term of A054743 that divides of n.

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