A359546 a(n) = 1 if there is no factor of the form p^p in n, but for some k-th arithmetic derivative (k >= 1) of n such a factor exists; otherwise 0.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1

Offset

1

Comments

Question: What can be said about the distribution of 0's and 1's in this sequence? Compare also to A328308, A341996 and A359543.

Links

Antti Karttunen, Table of n, a(n) for n = 1..100000

Index entries for characteristic functions

Formula

a(n) = A341999(n) - A342023(n).

a(n) = A359550(n) * A341999(n).

a(n) = [A256750(n) < 0], where [ ] is the Iverson bracket.

Example

a(15) = 1, because although 15 itself is not in A100716, its arithmetic derivative 15' = 8 is there.
a(26) = 1, as although neither 26 nor 26' = 15 are in A100716, the second derivative of 26, 26'' = 15' = 8 is there.

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));
A327936(n) = { my(f = factor(n)); for(k=1, #f~, f[k, 2] = (f[k, 2]>=f[k, 1])); factorback(f); };
A341999(n) = if(!n, n, while(n>1, n = A003415checked(n)); (!n));
A359546(n) = ((1==A327936(n))&&A341999(n));

Crossrefs

Characteristic function of A359547.

Cf. A003415, A048103, A256750, A327936, A328308, A341996, A341999, A342023, A359543, A359550.

Sequence in context: A359162 A327932 A373979 * A354989 A277161 A353481

Adjacent sequences: A359543 A359544 A359545 * A359547 A359548 A359549

Keyword

nonn

Author

Antti Karttunen, Jan 05 2023

Status

approved