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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

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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

(SageMath)

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.