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

%I A309391 #53 Feb 16 2025 08:33:55
%S A309391 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,
%T A309391 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,
%U A309391 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
%N A309391 a(n) = gcd(n, A064169(n-2)) for n > 2.
%C A309391 Probably, there are no composite terms in this sequence.
%C A309391 For n > 2, a(n) = gcd(n, A001008(n-1)).
%C A309391 By Wolstenholme's theorem, if p is an odd prime, then a(p) = p.
%C A309391 Conjecture: for n > 2, if a(n) = n, then n is a prime.
%C A309391 If so, then there are no pseudoprimes n such that a(n) = n.
%C A309391 Composite numbers m <> p^2 for which a(m) > 1 are 88, 1290, 9339, ...
%H A309391 Robert Israel, <a href="/A309391/b309391.txt">Table of n, a(n) for n = 3..10000</a>
%H A309391 Romeo Mestrovic, <a href="https://arxiv.org/abs/1111.3057">Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)</a>, arXiv:1111.3057 [math.NT], 2001.
%H A309391 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WolstenholmesTheorem.html">Wolstenholme's Theorem</a>.
%H A309391 Wikipedia, <a href="https://en.wikipedia.org/wiki/Wolstenholme%27s_theorem">Wolstenholme's theorem</a>.
%F A309391 a(p) = p for every odd prime p.
%F A309391 a(p^2) = p iff p > 3 is a prime.
%F A309391 Note that a(n) >= A089026(n) for n > 2.
%e A309391 a(25) = gcd(25, A064169(25-2)) = gcd(25, 325333835) = 5,
%e A309391 a(25) = gcd(25, A001008(25-1)) = gcd(25, 1347822955) = 5.
%e A309391 It should be noted that a(88) = 11, a(1290) = 43, a(9339) = 11, ...
%p A309391 H:= 0:
%p A309391 for n from 3 to 100 do
%p A309391   H:= H + 1/(n-2);
%p A309391   A[n]:= igcd(n, numer(H)-denom(H));
%p A309391 od:
%p A309391 seq(A[i],i=3..100); # _Robert Israel_, Aug 04 2019
%t A309391 a[n_] := GCD[n, Numerator[(h = HarmonicNumber[n-2])] - Denominator[h]]; Array[a, 81, 3]
%o A309391 (Magma) [Gcd(k, Numerator(a)-Denominator(a)) where a is HarmonicNumber(k-2):k in [3..90]]; // _Marius A. Burtea_, Jul 29 2019
%Y A309391 Cf. A001008, A002805, A007406 (see our comment), A064169, A065091, A089026, A309397.
%K A309391 nonn
%O A309391 3,1
%A A309391 _Amiram Eldar_ and _Thomas Ordowski_, Jul 28 2019