A322936 - OEIS

A322936

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

1, 0, 0, 0, 3, 0, 4, 5, 3, 5, 5, 7, 7, 3, 4, 6, 7, 8, 9, 5, 7, 5, 7, 8, 9, 10, 11, 3, 5, 9, 11, 4, 8, 11, 13, 7, 9, 11, 13, 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 5, 7, 11, 13, 4, 5, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 3, 7, 9, 11, 13, 17, 8, 11, 13, 16, 17, 19

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OFFSET

1,5

COMMENTS

a is strongly prime to n if and only if a <= n is prime to n and a 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..78.

Peter Luschny, Strong Coprimality

EXAMPLE

The length of row n is A181830(n) = phi(n) - tau(n-1). The triangular array starts:

[1] {1}

[2] {}

[3] {}

[4] {}

[5] {3}

[6] {}

[7] {4, 5}

[8] {3, 5}

[9] {5, 7}

[11] {3, 4, 6, 7, 8, 9}

[12] {5, 7}

[10] {7}

[13] {5, 7, 8, 9, 10, 11}

[14] {3, 5, 9, 11}

[15] {4, 8, 11, 13}

[16] {7, 9, 11, 13}

[17] {3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15}

[18] {5, 7, 11, 13}

[19] {4, 5, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17}

[20] {3, 7, 9, 11, 13, 17}

MAPLE

StrongCoprimes := n -> select(k -> igcd(k, n)=1, {$1..n}) minus numtheory:-divisors(n-1):

A322936row:= proc(n) if n in {2, 3, 4, 6} then return 0 else op(StrongCoprimes(n)) fi end:

seq(A322936row(n), n=1..20);

MATHEMATICA

Table[If[n == 1, {1}, Select[Range[2, n], And[GCD[#, n] == 1, Mod[n - 1, #] != 0] &] /. {} -> {0}], {n, 21}] // Flatten (* Michael De Vlieger, Apr 01 2019 *)

PROG

(Sage)

def primeto(n):

return [p for p in range(n) if gcd(p, n) == 1]

def strongly_primeto(n):

return [p for p in set(primeto(n)) - set((n-1).divisors())]

def A322936row(n):

if n == 1: return [1]

if n in [2, 3, 4, 6]: return [0]

return sorted(strongly_primeto(n))

for n in (1..21): print(A322936row(n))

CROSSREFS