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A234016 Partial sums of the characteristic function of A055938.

%I A234016 #22 Jul 16 2022 00:54:34
%S A234016 0,0,1,1,1,2,3,3,3,4,4,4,5,6,7,7,7,8,8,8,9,10,10,10,11,11,11,12,13,14,
%T A234016 15,15,15,16,16,16,17,18,18,18,19,19,19,20,21,22,22,22,23,23,23,24,25,
%U A234016 25,25,26,26,26,27,28,29,30,31,31,31,32,32,32,33,34
%N A234016 Partial sums of the characteristic function of A055938.
%C A234016 Also: a(0) = a(1) = 0, and thereafter, a(n) = the largest k such that A055938(k) <= n.
%C A234016 Conjecture: partial sums of A308187 (i.e, A308187 is the characteristic function of A055938). - _Sean A. Irvine_, Jul 16 2022
%H A234016 Antti Karttunen, <a href="/A234016/b234016.txt">Table of n, a(n) for n = 0..8191</a>
%F A234016 If n < 2, a(n)=0, otherwise a(n) = a(n-1) + (1-A079559(n)).
%F A234016 a(n) = n - (A046699(n+2)-1) [With A046699's October 2012 starting offset].
%o A234016 (Scheme, with memoizing definec-macro from _Antti Karttunen_'s IntSeq-library)
%o A234016 (definec (A234016 n) (if (< n 2) 0 (+ (A234016 (- n 1)) (- 1 (A079559 n)))))
%o A234016 ;; Alternatively, based on A046699, with its October 2012 starting offset:
%o A234016 (define (A234016 n) (- n (- (A046699 (+ n 2)) 1)))
%o A234016 (Python)
%o A234016 from sympy import factorial
%o A234016 def a046699(n):
%o A234016     if n<3: return 1
%o A234016     s=1
%o A234016     while factorial(2*s)%(2**(n - 1)): s+=1
%o A234016     return s
%o A234016 def a(n): return n - (a046699(n + 2) - 1)
%o A234016 print([a(n) for n in range(101)]) # _Indranil Ghosh_, Jun 11 2017
%Y A234016 Cf. A046699, A055938, A079559, A234017, A233275, A308187.
%K A234016 nonn
%O A234016 0,6
%A A234016 _Antti Karttunen_, Dec 18 2013