login
The exponentially odious part of n: the largest unitary divisor of n that is an exponentially odious number (A270428).
5

%I #15 Nov 23 2023 15:16:14

%S 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,

%T 1,28,29,30,31,1,33,34,35,36,37,38,39,5,41,42,43,44,45,46,47,48,49,50,

%U 51,52,53,2,55,7,57,58,59,60,61,62,63,1,65,66,67,68,69,70

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

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

%H Amiram Eldar, <a href="/A367514/b367514.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

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

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

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

%F 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)).

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

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

%o (Python)

%o from math import prod

%o from sympy import factorint

%o 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

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

%Y Cf. A056192, A270418.

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

%K nonn,easy,mult

%O 1,2

%A _Amiram Eldar_, Nov 21 2023