A309391 a(n) = gcd(n, A064169(n-2)) for n > 2.

3, 1, 5, 1, 7, 1, 1, 1, 11, 1, 13, 1, 1, 1, 17, 1, 19, 1, 1, 1, 23, 1, 5, 1, 1, 1, 29, 1, 31, 1, 1, 1, 1, 1, 37, 1, 1, 1, 41, 1, 43, 1, 1, 1, 47, 1, 7, 1, 1, 1, 53, 1, 1, 1, 1, 1, 59, 1, 61, 1, 1, 1, 1, 1, 67, 1, 1, 1, 71, 1, 73, 1, 1, 1, 1, 1, 79, 1, 1, 1, 83, 1, 1, 1, 1, 11, 89, 1

Offset: 3

Author

Amiram Eldar and Thomas Ordowski, Jul 28 2019

Keywords

• nonn

Comments

Probably, there are no composite terms in this sequence.

For n > 2, a(n) = gcd(n, A001008(n-1)).

By Wolstenholme's theorem, if p is an odd prime, then a(p) = p.

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

If so, then there are no pseudoprimes n such that a(n) = n.

Composite numbers m <> p^2 for which a(m) > 1 are 88, 1290, 9339, ...

Examples

			a(25) = gcd(25, A064169(25-2)) = gcd(25, 325333835) = 5,
a(25) = gcd(25, A001008(25-1)) = gcd(25, 1347822955) = 5.
It should be noted that a(88) = 11, a(1290) = 43, a(9339) = 11, ...
		

Links

• Robert Israel, Table of n, a(n) for n = 3..10000
• Romeo Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012), arXiv:1111.3057 [math.NT], 2001.
• Eric Weisstein's World of Mathematics, Wolstenholme's Theorem.
• Wikipedia, Wolstenholme's theorem.

Crossrefs

Cf. A001008, A002805, A007406 (see our comment), A064169, A065091, A089026, A309397.

Programs

Magma

[Gcd(k, Numerator(a)-Denominator(a)) where a is HarmonicNumber(k-2):k in [3..90]]; // Marius A. Burtea, Jul 29 2019

Maple

H:= 0:
for n from 3 to 100 do
  H:= H + 1/(n-2);
  A[n]:= igcd(n, numer(H)-denom(H));
od:
seq(A[i],i=3..100); # Robert Israel, Aug 04 2019

Mathematica

a[n_] := GCD[n, Numerator[(h = HarmonicNumber[n-2])] - Denominator[h]]; Array[a, 81, 3]

Formula

a(p) = p for every odd prime p.

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

Note that a(n) >= A089026(n) for n > 2.