A341999 a(n) = 1 if the k-th arithmetic derivative is nonzero for all k >= 0, otherwise 0.

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, 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, 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

Offset

0

Comments

Characteristic function of A099309.

Links

Antti Karttunen, Table of n, a(n) for n = 0..65537

Index entries for characteristic functions

Formula

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).

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

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

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

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

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

a(n) = 1 - A328308(n), and for n >= 1, a(n) = A342023(n) + A359546(n). - Antti Karttunen, Jan 05 2023

Prog

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

Crossrefs

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

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

Sequence in context: A144596 A188187 A341996 * A118685 A244063 A080343

Adjacent sequences: A341996 A341997 A341998 * A342000 A342001 A342002

Keyword

nonn

Author

Antti Karttunen, Feb 28 2021

Status

approved