A367514 The exponentially odious part of n: the largest unitary divisor of n that is an exponentially odious number (A270428).

1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 3, 25, 26, 1, 28, 29, 30, 31, 1, 33, 34, 35, 36, 37, 38, 39, 5, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 2, 55, 7, 57, 58, 59, 60, 61, 62, 63, 1, 65, 66, 67, 68, 69, 70

Offset: 1

Author

Amiram Eldar, Nov 21 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A056192 at n = 32, and from A270418 and A367168 at n = 128.

Links

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

Crossrefs

Cf. A000069, A001221, A010060, A034444, A102392, A262675, A270428, A293439, A366901, A366903, A367515.

Cf. A056192, A270418.

Similar sequences: A350388, A350389, A366905, A367168, A367513.

Programs

Mathematica

f[p_, e_] := p^(e*ThueMorse[e]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

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

Python

from math import prod
from sympy import factorint
def A367514(n): return prod(p**e for p, e in factorint(n).items() if e.bit_count()&1) # Chai Wah Wu, Nov 23 2023

Formula

Multiplicative with a(p^e) = p^(e*A010060(e)) = p^A102392(e).

a(n) = n/A367513(n).

A001221(a(n)) = A293439(n).

A034444(a(n)) = A367515(n).

a(n) >= 1, with equality if and only if n is an exponentially evil number (A262675).

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

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} f(1/p) = 0.88585652437242918295..., and f(x) = (x+2)/(2*(x+1)) + (x/2) * Product_{k>=0} (1 - x^(2^k)).