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A292376 a(1) = 0, a(2n) = 0, and for odd numbers n > 1, a(n) = a(A064989(n)) + [n == 3 (mod 4)].

%I A292376 #15 Sep 18 2017 09:25:44
%S A292376 0,0,1,0,1,0,2,0,0,0,3,0,3,0,1,0,3,0,4,0,0,0,5,0,0,0,1,0,5,0,6,0,0,0,
%T A292376 2,0,6,0,1,0,6,0,7,0,0,0,8,0,0,0,1,0,8,0,1,0,0,0,9,0,9,0,1,0,0,0,10,0,
%U A292376 0,0,11,0,11,0,1,0,2,0,12,0,0,0,13,0,1,0,1,0,13,0,2,0,0,0,2,0,13,0,1,0,13,0,14,0,0
%N A292376 a(1) = 0, a(2n) = 0, and for odd numbers n > 1, a(n) = a(A064989(n)) + [n == 3 (mod 4)].
%C A292376 For odd numbers > 1, iterate the map x -> A064989(x), which shifts every prime in the prime factorization of n one index step towards smaller primes. a(n) counts the numbers of the form 4k+3 encountered until the first number which is even has been reached. This count includes also n itself if it is of the form 4k+3 (A004767).
%C A292376 In other words, locate the position where n is in square array A246278 and moving up by that column, count all numbers of the form 4k+3 before an even number at the top of the column is reached.
%H A292376 Antti Karttunen, <a href="/A292376/b292376.txt">Table of n, a(n) for n = 1..16384</a>
%H A292376 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A292376 a(1) = 0, a(2n) = 0, and for odd numbers n > 1, a(n) = a(A064989(n)) + floor((n mod 4)/3).
%F A292376 Other identities and observations. For n >= 1.
%F A292376 a(n) <= A292377(n).
%F A292376 For n >= 2, a(n) + A292374(n) + 1 = A055396(n).
%t A292376 a[1] = 0; a[n_] := a[n] = If[EvenQ@ n, 0, a[Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n] + Boole[Mod[n, 4] == 3]]; Array[a, 105] (* _Michael De Vlieger_, Sep 17 2017 *)
%o A292376 (Scheme, with memoization-macro definec)
%o A292376 (definec (A292376 n) (if (or (even? n) (= 1 n)) 0 (+ (floor->exact (/ (modulo n 4) 3)) (A292376 (A064989 n)))))
%Y A292376 Cf. A055396, A064989, A246271, A246278, A292374, A292377.
%Y A292376 Cf. also A038802 (odd bisection of a(n) + A292374(n)).
%K A292376 nonn
%O A292376 1,7
%A A292376 _Antti Karttunen_, Sep 17 2017