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

0,6

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.

Links

Table of n, a(n) for n=0..50.

Peter Luschny, Strong coprimality.

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A321463 A197689 A201942 * A241536 A080407 A197335

Adjacent sequences: A181828 A181829 A181830 * A181832 A181833 A181834

Keyword

nonn

Author

Peter Luschny, Nov 17 2010

Extensions

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

Status

approved