A181830 The number of positive integers <= n that are strongly prime to n.

0, 0, 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

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.

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

Formula

a(n) = phi(n) - tau(n-1) for n > 1, where phi(n) = A000010(n) and tau(n) = A000005(n).

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.

Mathematica

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

Prog

(PARI) a(n)=if(n<2, 0, eulerphi(n)-numdiv(n-1));
for (i=0, 66, print1(a(i), ", ")) \\ Michel Marcus, May 22 2017
(SageMath)
def isstrongprimeto(k, n): return not(k.divides(n - 1)) and gcd(k, n) == 1
print([sum(int(isstrongprimeto(k, n)) for k in srange(n+1)) for n in srange(67)])
# Peter Luschny, Dec 03 2023

Crossrefs

Cf. A000005, A000010, A002522, A050384, A181831, A181832, A181833, A181834, A181835, A181836.

Sequence in context: A295691 A285183 A255399 * A352736 A086612 A128207

Adjacent sequences: A181827 A181828 A181829 * A181831 A181832 A181833

Keyword

nonn,easy

Author

Peter Luschny, Nov 17 2010

Extensions

Corrected a(1) to 0 by Peter Luschny, Dec 03 2023

Status

approved