A181831 The sum of positive integers <= n that are strongly prime to n.

0, 0, 0, 0, 0, 3, 0, 9, 8, 12, 7, 37, 12, 50, 28, 36, 40, 105, 36, 132, 60, 84, 78, 217, 72, 190, 125, 201, 128, 350, 90, 393, 224, 267, 224, 366, 168, 575, 304, 408, 264, 730, 210, 807, 396, 456, 428, 1009, 336, 905, 443

Offset: 0

Author

Peter Luschny, Nov 17 2010

Keywords

• nonn

Comments

k is strongly prime to n iff k is relatively prime to n and k does not divide n-1.

a(n) = A023896(n) - A000203(n-1) if n > 1 and a(n) = 0 for n = 0,1.

Examples

			a(11) = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 - 1 - 2 - 5 - 10 = 37.
		

Links

• Peter Luschny, Strong coprimality.

Crossrefs

Cf. A181830, A181832, A181833, A181834, A181835, A181836, A023896, A000203.

Programs

Maple

with(numtheory):
A181831 := n -> `if`(n<2,0,n*phi(n)/2-sigma(n-1)):

Mathematica

Join[{0,0},Table[Total[Select[Range[n],CoprimeQ[#,n]&&!Divisible[n-1,#]&]],{n,2,50}]] (* Harvey P. Dale, Apr 09 2013 *)

SageMath

def isstrongprimeto(k, n): return not(k.divides(n-1)) and gcd(k, n) == 1
def a(n): return sum(k for k in srange(n + 1) if isstrongprimeto(k, n))
print([a(n) for n in range(51)])
# Alternative:
def a(n): return 0 if n < 2 else n*euler_phi(n)//2 - sigma(n - 1, 1)
# Peter Luschny, Dec 03 2023

Extensions

a(0) corrected by Peter Luschny, Dec 03 2023