A367482 Number of nontrivial divisors of n whose arithmetic derivative is coprime to n.

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 3, 1, 1, 3, 1, 3, 3, 3, 1, 3, 1, 3, 1, 3, 1, 5, 1, 1, 3, 3, 3, 3, 1, 3, 3, 3, 1, 5, 1, 3, 3, 3, 1, 3, 1, 3, 3, 3, 1, 3, 3, 3, 3, 3, 1, 5, 1, 3, 3, 1, 3, 6, 1, 3, 3, 5, 1, 3, 1, 3, 3, 3, 3, 5, 1, 3, 1, 3, 1, 5, 3, 3, 3, 3, 1, 5, 3, 3, 3, 3, 3, 3, 1, 3, 3, 3, 1, 6, 1, 3, 5

Offset

1,6

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

a(n) = Sum_{d|n, d>1, gcd(n,d')=1} 1.

Example

a(66) = 6. The nontrivial divisors of 66 are {2, 3, 6, 11, 22, 33, 66} with derivatives {1, 1, 5, 1, 13, 14, 61}, 6 of which are coprime to n.

Mathematica

f[x_] := f[x] = If[x < 2, 0, x Total[#2/#1 & @@@ FactorInteger[x]]]; {0}~Join~Table[DivisorSum[n, Boole[CoprimeQ[f[#], n]] &], {n, 2, 104}] (* Michael De Vlieger, Nov 24 2023 *)

Prog

(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A367482(n) = sumdiv(n, d, (d>1)&&(1==gcd(n, A003415(d)))); \\ Antti Karttunen, Jan 13 2025

Crossrefs

Cf. A003415 (n'), A367483.

Sequence in context: A132468 A353235 A243915 * A367095 A309307 A325446

Adjacent sequences: A367479 A367480 A367481 * A367483 A367484 A367485

Keyword

nonn

Author

Wesley Ivan Hurt, Nov 24 2023

Extensions

More terms from Antti Karttunen, Jan 13 2025

Status

approved