A087625 Number of primes in the ring Z_n.

0, 0, 0, 1, 0, 3, 0, 2, 2, 5, 0, 4, 0, 7, 6, 4, 0, 8, 0, 6, 8, 11, 0, 8, 4, 13, 6, 8, 0, 14, 0, 8, 12, 17, 10, 10, 0, 19, 14, 12, 0, 20, 0, 12, 14, 23, 0, 16, 6, 24, 18, 14, 0, 24, 14, 16, 20, 29, 0, 20, 0, 31, 18, 16, 16, 32, 0, 18, 24, 34, 0, 20, 0, 37, 28, 20, 16, 38, 0, 24, 18, 41, 0, 28, 20, 43, 30, 24, 0, 38, 18, 24, 32, 47, 22, 32, 0, 48

Offset

1,6

Comments

a(n) <= n-phi(n)-1.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

a(n)=Sum'_{p|n} A087623(p, n), where the sum is over all primes p < n, p | n.

a(p)=0 if p prime.

a(p^k)=p^{k-2}(p-1) if p prime, k>=2.

a(p^k q)=p^{k-2}(p-1)(p+q-1) if p, q primes (q!=p), k>=2.

a(pq)=p+q-2 if p, q primes, p!=q.

A(p^k q^h)=p^{k-2}q^{h-2}(p-1)(q-1)(p+q) if p, q primes (q!=p),

Example

a(6)=3 because the three primes in Z_6 are 2,3,4, being 2 and 4 associates. a(500)=5(2-1)(5-1)(2+5)=140.

Prog

(PARI) A087625(n) = sumdiv(n, p, if((p<n)&&isprime(p), sum(k=1, n-1, (gcd(k, n)==p)), 0)); \\ Antti Karttunen, Mar 04 2018

Crossrefs

Cf. A000010, A087623-A087624.

Sequence in context: A285006 A369747 A300722 * A300720 A371981 A280238

Adjacent sequences: A087622 A087623 A087624 * A087626 A087627 A087628

Keyword

nonn,easy

Author

Michele Dondi (bik.mido(AT)tiscalinet.it), Sep 14 2003

Extensions

More terms from Antti Karttunen, Mar 04 2018

Status

approved