A393041 - OEIS

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A393041

Number of trailing zeros in the primorial base representation of the arithmetic derivative of n.

5

0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 2, 3, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 2, 1, 0, 0, 2, 0, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 2, 1, 0, 2, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1

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OFFSET

2,7

COMMENTS

A002110(a(n)) is the largest primorial that divides A003415(n).

LINKS

Antti Karttunen, Table of n, a(n) for n = 2..12000

Index entries for sequences related to primorial base.

FORMULA

a(n) = A276084(A003415(n)).

For n >= 0, a(A393042(n)) = n.

EXAMPLE

A003415(2) = 1, and A049345(1) = 1, with no (zero) trailing 0's, therefore a(2) = 0.

A003415(36) = 60, and A049345(60) = 2000, with three trailing 0's, therefore a(36) = 3.

PROG

(PARI)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

A276084(n) = { for(i=1, oo, if(n%prime(i), return(i-1))); }

A393041(n) = A276084(A003415(n));

CROSSREFS

Cf. A002110, A003415, A049345, A276084.

Positions of terms 0..3 in this sequence are given by A235991 (0), A393043 (1), A393045 (2), A393047 (3).

Positions of nonzero terms are given by A235992 after its two initial terms.

Index of the first occurrence of n is given by A393042, that most likely are also positions of the records.

Sequence in context: A380955 A058190 A055736 * A006997 A141612 A316342

Adjacent sequences: A393038 A393039 A393040 * A393042 A393043 A393044

KEYWORD

nonn,base,easy

AUTHOR

Antti Karttunen, Feb 02 2026

STATUS

approved