A309397 a(n) = gcd(n^2, A001008(n-1)) for n > 1.

1, 3, 1, 25, 1, 49, 1, 1, 1, 121, 1, 169, 1, 1, 1, 289, 1, 361, 1, 1, 1, 529, 1, 5, 1, 1, 1, 841, 1, 961, 1, 1, 1, 1, 1, 1369, 1, 1, 1, 1681, 1, 1849, 1, 1, 1, 2209, 1, 7, 1, 1, 1, 2809, 1, 1, 1, 1, 1, 3481, 1, 3721, 1, 1, 1, 1, 1, 4489, 1, 1, 1, 5041, 1, 5329

Offset: 2

Author

Amiram Eldar and Thomas Ordowski, Jul 28 2019

Keywords

• nonn

Comments

By Wolstenholme's theorem, if p > 3 is prime, then a(p) = p^2.

Conjecture: for n > 3, if a(n) = n^2, then n is a prime.

Note: the weak pseudoprimes n such that a(n) = n are not known.

Composite numbers m <> p^2 for which a(m) > 1 are the same as in A309391: 88, 1290, 9339, ...

Examples

			a(11) = gcd(11^2, A001008(11-1)) = gcd(121, 7381) = 121.
		

Links

• R. Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012), arXiv:1111.3057 [math.NT], 2011.
• Eric Weisstein's World of Mathematics, Wolstenholme's Theorem.
• Wikipedia, Wolstenholme's theorem.

Crossrefs

Cf. A001008, A007406 (see our comment), A309391.

Programs

Magma

[Gcd(k^2, Numerator(HarmonicNumber(k-1))):k in [2..80]]; // Marius A. Burtea, Jul 28 2019

Mathematica

a[n_] := GCD[n^2, Numerator[HarmonicNumber[n-1]]]; Array[a, 72, 2]

Python

from sympy import gcd, harmonic
def A309387(n):
    return gcd(n**2,harmonic(n-1).p) # Chai Wah Wu, Jul 31 2019

Formula

a(n) = A309391(n) for composite n.

a(p) = p^2 for every prime p > 3.

a(p^2) = p iff p > 3 is a prime.