A341999 - OEIS

%I #19 Jan 05 2023 18:30:47

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

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

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

%N a(n) = 1 if the k-th arithmetic derivative is nonzero for all k >= 0, otherwise 0.

%C Characteristic function of A099309.

%H Antti Karttunen, <a href="/A341999/b341999.txt">Table of n, a(n) for n = 0..65537</a>

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

%F a(n) = 1 if n is in A100716 or ends there by repeated applications of A003415, otherwise a(n) = 0 (when n instead reaches 0 by such iteration).

%F For all n, a(n) >= A341996(n).

%F For all n > 0, a(A099309(n)) = a(A100716(n)) = 1.

%F For all n > 0, a(n) = [A256750(n) < 1].

%F For all n > 0, a(n) >= [A129251(n)>0], i.e., if A129251(n) is nonzero, then certainly a(n) = 1.

%F For all n > 1, a(n) >= [A341997(n) > 1].

%F a(n) = 1 - A328308(n), and for n >= 1, a(n) = A342023(n) + A359546(n). - _Antti Karttunen_, Jan 05 2023

%o (PARI)

%o A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i,2]>=f[i,1],return(0), s += f[i, 2]/f[i, 1])); (n*s));

%o A341999(n) = if(!n,n,while(n>1, n = A003415checked(n)); (!n));

%Y Cf. A003415, A100716, A129251, A256750, A341996, A341997.

%Y Cf. A099308 (positions of zeros), A099309 (of ones), A328308 (one's complement), A342023, A359542 (inverse Möbius transform), A359546.

%K nonn

%O 0

%A _Antti Karttunen_, Feb 28 2021