A368915 - OEIS

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A368915

a(n) = 1 if there is no prime p such that p^p divides the arithmetic derivative of n, and 0 otherwise.

0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1`
	COMMENTS	`Question: What is the asymptotic mean of this sequence (and its complement A341996)? Knowing the value for A360111 would solve this. See also related sequences like A354874 and A368916.`
	LINKS	`Antti Karttunen, Table of n, a(n) for n = 1..100000` `Index entries for characteristic functions`
	FORMULA	`a(1) = 0; for n > 1, a(n) = A359550(A003415(n)).` `For all n > 1, a(n) = 1 - A341996(n) = A359550(n) - A360111(n).` `For all n > 1, A359550(n) >= a(n) >= A328308(n).` `For all n >= 1, a(n) >= A354874(n).` `a(n) = A368914(n) - A368913(n).`
	PROG	`(PARI)` `A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));` `A359550(n) = { my(f = factor(n)); prod(k=1, #f~, (f[k, 2]<f[k, 1])); };` `A368915(n) = ((n>1)&&A359550(A003415(n)));`
	CROSSREFS	`Characteristic function of A358215.` `Cf. A003415, A328308, A341996 (one's complement), A354874, A359550, A360111, A368913, A368914, A368916 [= a(A276086(n))].` `Sequence in context: A284929 A286746 A155897 * A144610 A188068 A181632` `Adjacent sequences: A368912 A368913 A368914 * A368916 A368917 A368918`
	KEYWORD	`nonn`
	AUTHOR	`Antti Karttunen, Jan 09 2024`
	STATUS	`approved`

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Last modified August 18 13:06 EDT 2024. Contains 375269 sequences. (Running on oeis4.)