A322937 Triangular array in which the n-th row lists the primes strongly prime to n (in ascending order). For the empty rows n = 1, 2, 3, 4 and 6 we set by convention 0.

0, 0, 0, 0, 3, 0, 5, 3, 5, 5, 7, 7, 3, 7, 5, 7, 5, 7, 11, 3, 5, 11, 11, 13, 7, 11, 13, 3, 5, 7, 11, 13, 5, 7, 11, 13, 5, 7, 11, 13, 17, 3, 7, 11, 13, 17, 11, 13, 17, 19, 5, 13, 17, 19, 3, 5, 7, 13, 17, 19, 5, 7, 11, 13, 17, 19, 7, 11, 13, 17, 19, 23

Offset

1,5

Comments

A number k is strongly prime to n if and only if k <= n is prime to n and k does not divide n-1. See the link to 'Strong Coprimality'. (Our terminology follows the plea of Knuth, Graham and Patashnik in Concrete Mathematics, p. 115.)

Links

Table of n, a(n) for n=1..72.

Peter Luschny, Strong Coprimality

Example

The length of row n is A181834(n). The triangular array starts:
[1] {}
[2] {}
[3] {}
[4] {}
[5] {3}
[6] {}
[7] {5}
[8] {3, 5}
[9] {5, 7}
[10] {7}
[11] {3, 7}
[12] {5, 7}
[13] {5, 7, 11}
[14] {3, 5, 11}
[15] {11, 13}
[16] {7, 11, 13}
[17] {3, 5, 7, 11, 13}
[18] {5, 7, 11, 13}
[19] {5, 7, 11, 13, 17}
[20] {3, 7, 11, 13, 17}

Maple

Primes := n -> select(isprime, {$1..n}):
StrongCoprimes := n -> select(k->igcd(k, n)=1, {$1..n}) minus numtheory:-divisors(n-1):
StrongCoprimePrimes := n -> Primes(n) intersect StrongCoprimes(n):
A322937row := proc(n) if n in {1, 2, 3, 4, 6} then return 0 else op(StrongCoprimePrimes(n)) fi end:
seq(A322937row(n), n=1..25);

Mathematica

Table[Select[Prime@ Range@ PrimePi@ n, And[GCD[#, n] == 1, Mod[n - 1, #] != 0] &] /. {} -> {0}, {n, 25}] // Flatten (* Michael De Vlieger, Apr 01 2019 *)

Prog

(Sage)
def primes_primeto(n):
    return [p for p in prime_range(n) if gcd(p, n) == 1]
def primes_strongly_primeto(n):
    return [p for p in set(primes_primeto(n)) - set((n-1).divisors())]
def A322937row(n):
    if n in [1, 2, 3, 4, 6]: return [0]
    return sorted(primes_strongly_primeto(n))
for n in (1..25): print(A322937row(n))

Crossrefs

Cf. A000010, A038566, A181831, A181832, A181833, A181834, A181835, A181836, A322936.

Sequence in context: A179179 A291503 A108500 * A326989 A326937 A336597

Adjacent sequences: A322934 A322935 A322936 * A322938 A322939 A322940

Keyword

nonn,tabf

Author

Peter Luschny, Apr 01 2019

Status

approved