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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.
7
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.
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.
Sequence in context: A359162 A327932 A373979 * A354989 A277161 A353481
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jan 05 2023
STATUS
approved