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A181830 The number of positive integers <= n that are strongly prime to n. 13
0, 1, 0, 0, 0, 1, 0, 2, 2, 2, 1, 6, 2, 6, 4, 4, 4, 11, 4, 12, 6, 6, 6, 18, 6, 12, 9, 14, 8, 22, 6, 22, 14, 14, 12, 20, 8, 27, 16, 20, 12, 32, 10, 34, 18, 18, 16, 42, 14, 32, 17, 26, 20, 46, 16, 32, 20, 28, 24, 54, 14, 48, 28, 32, 26, 41, 16 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

k is strongly prime to n if and only if k is relatively prime to n and k does not divide n - 1.

a(n) = phi(n) - tau(n-1) if n > 0 and a(0) = 0. Here phi(n) = A000010(n) and tau(n) = A000005(n).

It is conjectured (see Scroggs link) that a(n) is also the number of cardboard braids that work with n slots. - Matthew Scroggs, Sep 23 2017

a(n) is odd if and only if n is in A002522 but n <> 2. - Robert Israel, Jun 20 2018

LINKS

Robert Israel, Table of n, a(n) for n = 0..10000

Peter Luschny, Strong coprimality

Matthew Scroggs, Braiding, pt. 2. Two results and a conjecture

EXAMPLE

a(11) = card({1,2,3,4,5,6,7,8,9,10} - {1,2,5,10}) = card({3,4,6,7,8,9}) = 6.

MAPLE

with(numtheory): A181830 := n -> `if`(n=0, 0, phi(n)-tau(n-1)):

StrongCoprimes := n -> select(k -> igcd(k, n)=1, {$1..n}) minus divisors(n-1):

A181830a := n -> nops(StrongCoprimes(n)):

MATHEMATICA

a[0]=0; a[1]=1; a[n_ /; n > 1] := Select[Range[n], CoprimeQ[#, n] && !Divisible[n-1, #] &] // Length; Table[a[n], {n, 0, 66}] (* Jean-Fran├žois Alcover, Jun 26 2013 *)

PROG

(Pari) a(n)=if(n<2, n, eulerphi(n)-numdiv(n-1));

for (i=0, 66, print1(a(i), ", ")) \\ Michel Marcus, May 22 2017

CROSSREFS

Cf. A000005, A000010, A002522, A181831, A181832, A181833, A181834, A181835, A181836.

Cf. A050384.

Sequence in context: A295691 A285183 A255399 * A086612 A128207 A306707

Adjacent sequences:  A181827 A181828 A181829 * A181831 A181832 A181833

KEYWORD

nonn,easy

AUTHOR

Peter Luschny, Nov 17 2010

STATUS

approved

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Last modified May 28 01:48 EDT 2020. Contains 334671 sequences. (Running on oeis4.)