A366905 The largest exponentially odious divisor of n.

1, 2, 3, 4, 5, 6, 7, 4, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 25, 26, 9, 28, 29, 30, 31, 16, 33, 34, 35, 36, 37, 38, 39, 20, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 18, 55, 28, 57, 58, 59, 60, 61, 62, 63, 16, 65, 66, 67, 68

Offset: 1

Author

Amiram Eldar, Oct 27 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A353897 at n = 128.

The largest divisor of n that is an exponentially odious number (A270428).

The number of exponentially odious divisors of n is A366901(n) and their sum is A366903(n).

Links

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

Crossrefs

Cf. A270428, A366901,A366903.

Similar sequences: A353897, A365683, A366906.

Programs

Mathematica

maxOdious[e_] := Module[{k = e}, While[EvenQ[DigitCount[k, 2, 1]], k--]; k]; f[p_, e_] := p^maxOdious[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

s(n) = {my(k = n); while(!(hammingweight(k)%2), k--); k;}
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^s(f[i, 2]));}

Formula

Multiplicative with a(p^e) = p^max{k=1..e, k odious}.

a(n) <= n, with equality if and only if n is exponentially odious number (A270428).

Sum_{k=1..n} a(k) ~ c*n^2, where c = (1/2) * Product_{p prime} (1 + Sum_{e>=1} (p^f(e) - p^(f(e-1)+1))/p^(2*e)) = 0.4636829525..., f(e) = max{k=1..e, k odious} for e >= 1, and f(0) = 0.