A356192 a(n) is the smallest cubefull exponentially odd number (A335988) that is divisible by n.

1, 8, 27, 8, 125, 216, 343, 8, 27, 1000, 1331, 216, 2197, 2744, 3375, 32, 4913, 216, 6859, 1000, 9261, 10648, 12167, 216, 125, 17576, 27, 2744, 24389, 27000, 29791, 32, 35937, 39304, 42875, 216, 50653, 54872, 59319, 1000, 68921, 74088, 79507, 10648, 3375, 97336

Offset: 1

Author

Amiram Eldar, Jul 29 2022

Keywords

• nonn
• easy
• mult

Comments

First differs from A053149 and A356193 at n=16.

Links

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

Crossrefs

Cf. A001694, A013664, A268335, A335988.

Similar sequences: A053149, A053143, A053167, A066638, A087320, A087321, A197863, A356191, A356193, A356194.

Programs

Mathematica

f[p_, e_] := If[OddQ[e], p^Max[e, 3], p^(e + 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50]

PARI

a(n) = {my(f=factor(n)); prod(i=1, #f~, if(f[i,2]%2, f[i,1]^max(f[i,2],3), f[i,1]^(f[i,2]+1)))};

Formula

Multiplicative with a(p^e) = p^max(e,3) if e is odd and p^(e+1) otherwise.

a(n) = n iff n is in A335988.

a(n) = A356191(n) iff n is a powerful number (A001694).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + (3*p^2-1)/(p^3*(p^2-1))) = 1.69824776889117043774... .

Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(6)/4) * Product_{p prime} (1 - 1/p^2 + 1/p^5 - 2/p^6 + 1/p^8 + 1/p^9 - 1/p^10) = 0.1559368144... . - Amiram Eldar, Nov 13 2022