1,5

The zeros of this sequence are listed in A048597, which is a finite sequence with 30 as its largest element. The proof, discovered independently by Schatunowsky and Wolfskehl, is elementary but relies on results that are not elementary, such as Bertrand's postulate.

However, one can get surprisingly far trying to prove it by elementary means relying only on elementary results. It is almost trivially easy to prove that a(n) is positive for any n > 30 that is not divisible by 6. In the odd case, a(n) is at least 4 + 8 + 16 = 28, and in the even case a(n) is at least 9 + 18 = 27. And if n is divisible by 6 and of the form 6p, where p is a prime greater than 5, then a(n) is at least 25. Using Bertrand's postulate seems to be a requirement for proving the theorem for primorials, near primorials and other multiples of 6 having more than three prime factors.

ProofWiki, Schatunowsky's theorem

For odd n, a(n) >= A003817(n) - 3. The hedge "or equal to" is needed for n = 3, 5 or 9.

a(n) = A023896(n) - A034387(n) + A008472(n) - 1. - Chai Wah Wu, Dec 22 2025

a(7) = 10 because the numbers 1 to 6 are all coprime to 7 and of those 4 and 6 are composite, they add up to 10.

a(8) = 0 because although 1, 3, 5 and 7 are coprime to 8, none of those numbers are composite.

Table[Plus@@Select[Range[2, n - 1], GCD[#, n] == 1 && Not[PrimeQ[#]] &], {n, 80}]

(Scala) def isPrime(num: Int): Boolean = Math.abs(num) match {

case 0 => false

case 1 => false

case n => (2 to Math.floor(Math.sqrt(n)).toInt) forall (p => n % p != 0)

}

def euclGCD(a: Int, b: Int): Int = b match {

case 0 => a

case n => Math.abs(euclGCD(b, a % b))

}

def addUpCompositeCoprimes(n: Int): Int = {

(2 to (n - 1)).filter(m => !isPrime(m) && euclGCD(m, n) == 1).sum

}

(1 to 100).map(addUpCompositeCoprimes)

Cf. A000203, A001065, A036997.

Cf. A325284, near primorials; A070826, half primorials.

Cf. A064819, difference between the square of the smallest prime not a divisor of primorial(n) and primorial(n) itself.

Cf. A023896, A034387, A008472.

Sequence in context: A174381 A184363 A331451 * A390040 A164735 A390039

Adjacent sequences: A390320 A390321 A390322 * A390324 A390325 A390326

nonn,easy

Alonso del Arte, Nov 01 2025

approved