A380085 The largest unitary divisor of n that is a term in A276078.

1, 2, 3, 1, 5, 6, 7, 1, 9, 10, 11, 3, 13, 14, 15, 1, 17, 18, 19, 5, 21, 22, 23, 3, 25, 26, 1, 7, 29, 30, 31, 1, 33, 34, 35, 9, 37, 38, 39, 5, 41, 42, 43, 11, 45, 46, 47, 3, 49, 50, 51, 13, 53, 2, 55, 7, 57, 58, 59, 15, 61, 62, 63, 1, 65, 66, 67, 17, 69, 70, 71

Offset: 1

Author

Amiram Eldar, Jan 11 2025

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

Cf. A000720, A077610, A276078, A325127, A377515, A380086, A380087, A380088.

Programs

Mathematica

f[p_, e_] := p^If[e <= PrimePi[p], e, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^if(f[i,2] <= primepi(f[i,1]), f[i,2], 0));}

Formula

Multiplicative with a(p^e) = p^e if e <= pi(p) = A000720(p), and 1 otherwise.

a(n) = 1 if and only if n is in A325127.

a(n) = n if and only if n is in A276078.

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{k>=1} (p(k)^(2*(k+1)) + p(k)^(2*k+1) - p(k)^(k+1) - p(k)^k + 1)/(p(k)^(2*k+1) * (p(k)+1)) = 0.76189494803691349595..., where p(k) = prime(k).