A324903 - OEIS

%I #30 May 11 2023 16:57:00

%S 0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,1,1,0,1,0,

%T 1,0,1,1,1,0,1,1,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,1,1,0,1,1,1,0,

%U 1,1,1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,1,1,0,1,0,1,1,1,1,1,0,1,0,1,0,1,1,1,0,1

%N a(n) = 1 if A007814(sigma(n)) > A007814(n), 0 otherwise. Here A007814(n) gives the 2-adic valuation of n.

%C Characteristic function of A216782.

%C Difference between 2-adic valuations of (2*n OR sigma(n)) and (n OR (sigma(n)-n)), where OR is bitwise-OR, A003986.

%H Antti Karttunen, <a href="/A324903/b324903.txt">Table of n, a(n) for n = 1..100000</a>

%H Jon Maiga, <a href="http://sequencedb.net/s/A324903">Computer-generated formulas for A324903</a>, Sequence Machine

%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%F a(n) = A324904(n) - A324902(n).

%F a(n) = A324883(A005940(1+n)).

%F From _Antti Karttunen_, Mar 20 2023: (Start)

%F These are also listed by Maiga's Sequence Machine:

%F a(n) = A059841(A098987(n)).

%F a(n) = A361023(n) - A361024(n) = A361023(2*n).

%F a(n) = A059841(A017665(n)) = A059841(sigma(n)/gcd(n, sigma(n))).

%F a(n) = A324902(2*n) - A324902(n).

%F a(n) = A000035(A249670(n)-A017666(n)) = A000035(A249670(n)+A017666(n)).

%F a(n) = -1 + A325636(n)/A009194(n).

%F a(n) = A169813(A325635(n)/A325634(n)).

%F (End)

%o (PARI) A324903(n) = (valuation(n,2)<valuation(sigma(n),2));

%o (PARI)

%o A318456(n) = bitor(n,sigma(n)-n);

%o A318466(n) = bitor(2*n, sigma(n));

%o A324902(n) = valuation(A318456(n),2);

%o A324904(n) = valuation(A318466(n),2);

%o A324903(n) = (A324904(n)-A324902(n));

%o (Python)

%o from sympy import divisor_sigma

%o def A324903(n): return int((~(m:=int(divisor_sigma(n)))&m-1).bit_length()>(~n&n-1).bit_length()) # _Chai Wah Wu_, Jul 11 2022

%Y Cf. A000035, A000203, A003986, A005940, A007814, A009194, A017665, A059841, A098987, A216782, A243473, A249670, A286357, A318456, A318466, A324883, A324902, A324904, A325636, A361023, A361024.

%K nonn

%O 1

%A _Antti Karttunen_, Mar 28 2019