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

1,2

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

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)).

Mathematica

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

Prog

(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

Crossrefs

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

Cf. A056192, A270418.

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

Sequence in context: A368171 A050985 A367168 * A270418 A056192 A255693

Adjacent sequences: A367511 A367512 A367513 * A367515 A367516 A367517

Keyword

nonn,easy,mult

Author

Amiram Eldar, Nov 21 2023

Status

approved