A342414 a(n) = A003415(n) / gcd(phi(n),A003415(n)), where A003415(n) is the arithmetic derivative of n, and phi is Euler totient function.

0, 1, 1, 2, 1, 5, 1, 3, 1, 7, 1, 4, 1, 3, 1, 4, 1, 7, 1, 3, 5, 13, 1, 11, 1, 5, 3, 8, 1, 31, 1, 5, 7, 19, 1, 5, 1, 7, 2, 17, 1, 41, 1, 12, 13, 25, 1, 7, 1, 9, 5, 7, 1, 9, 2, 23, 11, 31, 1, 23, 1, 11, 17, 6, 3, 61, 1, 9, 13, 59, 1, 13, 1, 13, 11, 20, 3, 71, 1, 11, 2, 43, 1, 31, 11, 15, 4, 7, 1, 41, 5, 24, 17, 49, 1, 17

Offset

1,4

Links

Antti Karttunen, Table of n, a(n) for n = 1..12500

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Formula

a(n) = A003415(n) / A342413(n) = A003415(n) / gcd(A000010(n),A003415(n)).

a(n) = A342001(n) / A342416(n).

Mathematica

Array[#1/GCD[#1, #2] & @@ {If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &@ Abs[#], EulerPhi[#]} &, 96] (* Michael De Vlieger, Mar 11 2021 *)

Prog

(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342414(n) = { my(u=A003415(n)); (u/gcd(eulerphi(n), u)); };

Crossrefs

Cf. A000010, A003415, A342001, A342008 (positions of ones), A342413, A342415, A342416.

Sequence in context: A047818 A378783 A055972 * A374964 A079168 A332085

Adjacent sequences: A342411 A342412 A342413 * A342415 A342416 A342417

Keyword

nonn

Author

Antti Karttunen, Mar 11 2021

Status

approved