A048864 Number of nonprime numbers (composites and 1) in the reduced residue system of n.

1, 1, 1, 1, 2, 1, 3, 1, 3, 2, 6, 1, 7, 2, 4, 3, 10, 1, 11, 2, 6, 4, 14, 1, 12, 5, 10, 5, 19, 1, 20, 6, 11, 7, 15, 3, 25, 8, 14, 6, 28, 2, 29, 8, 12, 10, 32, 3, 28, 7, 19, 11, 37, 4, 26, 10, 22, 14, 42, 2, 43, 14, 20, 15, 32, 5, 48, 15, 27, 8, 51, 6, 52, 17, 21, 17, 41, 6, 57, 12, 33, 20

Offset

1,5

Comments

Differs from A039776 at n = 20, 21, ...

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Abhijit A J, A. Satyanarayana Reddy, Number of non-primes in the set of units modulo n, arXiv:1907.09908 [math.GM], 2019.

Abhijit A. J. and A. Satyanarayana Reddy, Number of non-primes in the set of units modulo n, The Mathematics Student, Vol. 88, No. 1-2 (2019), 147-152.

Formula

a(n) = A036997(n) + 1. - Peter Luschny, Oct 22 2010

a(n) = A000010(n) - (A000720(n) - A001221(n)).

Example

At n = 10, we see that the numbers below 10 coprime to 10 are 1, 3, 7, 9. Removing 3 and 7, which are prime, we are left with two numbers, 1 and 9. Hence a(10) = 2.
At n = 100, phi(100) = 40, phi(100) - (pi(100) - A001221(100)) = 17, thus a(100) = 17.

Maple

A048864 := n -> nops(select(k->gcd(k, n)=1, remove(isprime, [$1..n]))); # Peter Luschny, Oct 22 2010

Mathematica

Array[EulerPhi@ # - (PrimePi@ # - PrimeNu@ #) &, 82] (* Michael De Vlieger, Jul 03 2016 *)
Table[Length[Select[Range[n], GCD[n, #] == 1 && Not[PrimeQ[#]] &]], {n, 80}] (* Alonso del Arte, Oct 02 2017 *)

Prog

(PARI) a(n) = eulerphi(n) - (primepi(n) - omega(n)); \\ Indranil Ghosh, Apr 27 2017
(Python)
from sympy import totient, primepi, primefactors
def a(n): return totient(n) - (primepi(n) - len(primefactors(n))) # Indranil Ghosh, Apr 27 2017

Crossrefs

Cf. A039776, A000010, A000720, A001221, A037228, A072022, A072023, A074915.

Sequence in context: A230070 A224762 A039776 * A379315 A003139 A349918

Adjacent sequences: A048861 A048862 A048863 * A048865 A048866 A048867

Keyword

nonn

Author

Labos Elemer

Extensions

Converted second formula to an equation, added commas to the example - R. J. Mathar, Oct 23 2010

Status

approved