# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a353235 Showing 1-1 of 1 %I A353235 #24 Jun 29 2024 22:52:10 %S A353235 0,1,1,1,1,3,1,1,1,3,1,3,1,3,2,1,1,4,1,3,2,3,1,3,1,3,2,3,1,6,1,1,2,3, %T A353235 2,4,1,3,2,3,1,6,1,3,3,3,1,3,1,4,2,3,1,6,2,3,2,3,1,6,1,3,3,1,2,6,1,3, %U A353235 2,6,1,4,1,3,3,3,2,6,1,3,2,3,1,6,2,3,2,3,1,9,2,3,2,3 %N A353235 Number of divisors of n whose arithmetic derivative is odd. %C A353235 Inverse Möbius transform of (n' mod 2), where n' is the arithmetic derivative of n (A003415). - _Wesley Ivan Hurt_, Jun 29 2024 %H A353235 Robert Israel, Table of n, a(n) for n = 1..10000 %F A353235 a(n) = Sum_{d|n} ((d') mod 2). %F A353235 a(n) = tau(n)/2 - (1/2) * Sum_{d|n} (-1)^(d'). %F A353235 a(n) = A000005(n) - A353236(n). %F A353235 a(n) = A000005(n)/2 - A353237(n)/2. %F A353235 From _Robert Israel_, Jun 05 2023: (Start) %F A353235 If n = 2^k * m where m is odd and k >= 1, a(n) = a(m) + A000005(m). %F A353235 If n is odd and squarefree, a(n) = 2^(A001222(n)-1). %F A353235 If p is an odd prime, a(p^k) = ceil(k/2). %F A353235 If k and m are odd, a(k*m) = A000005(k)*a(m) + A000005(m)*a(k) - 2*a(m)*a(k). %F A353235 (End) %e A353235 a(12) = 3; 12 has 3 divisors whose arithmetic derivatives are odd: 2' = 1, 3' = 1, and 6' = 5. %p A353235 aderodd:= proc(n) local t; option remember; %p A353235 (n*add(t[2]/t[1],t=ifactors(n)[2]))::odd %p A353235 end proc: %p A353235 f:= proc(n) local t; %p A353235 nops(select(aderodd, numtheory:-divisors(n))) %p A353235 end proc: %p A353235 map(f, [$1..100]); # _Robert Israel_, Jun 05 2023 %t A353235 d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := DivisorSum[n, 1 &, OddQ[d[#]] &]; Array[a, 100] (* _Amiram Eldar_, May 02 2022 *) %o A353235 (PARI) ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415 %o A353235 a(n) = sumdiv(n, d, ad(d) % 2); \\ _Michel Marcus_, May 02 2022 %Y A353235 Cf. A000005 (tau), A003415 (n'), A353236, A353237. %K A353235 nonn %O A353235 1,6 %A A353235 _Wesley Ivan Hurt_, May 01 2022 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE