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

%I A309397 #30 Feb 16 2025 08:33:55
%S A309397 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,
%T A309397 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,
%U A309397 1,1,1,1,1,3481,1,3721,1,1,1,1,1,4489,1,1,1,5041,1,5329
%N A309397 a(n) = gcd(n^2, A001008(n-1)) for n > 1.
%C A309397 By Wolstenholme's theorem, if p > 3 is prime, then a(p) = p^2.
%C A309397 Conjecture: for n > 3, if a(n) = n^2, then n is a prime.
%C A309397 Note: the weak pseudoprimes n such that a(n) = n are not known.
%C A309397 Composite numbers m <> p^2 for which a(m) > 1 are the same as in A309391: 88, 1290, 9339, ...
%H A309397 R. Meštrović, <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], 2011.
%H A309397 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WolstenholmesTheorem.html">Wolstenholme's Theorem</a>.
%H A309397 Wikipedia, <a href="https://en.wikipedia.org/wiki/Wolstenholme%27s_theorem">Wolstenholme's theorem</a>.
%F A309397 a(n) = A309391(n) for composite n.
%F A309397 a(p) = p^2 for every prime p > 3.
%F A309397 a(p^2) = p iff p > 3 is a prime.
%e A309397 a(11) = gcd(11^2, A001008(11-1)) = gcd(121, 7381) = 121.
%t A309397 a[n_] := GCD[n^2, Numerator[HarmonicNumber[n-1]]]; Array[a, 72, 2]
%o A309397 (Magma) [Gcd(k^2, Numerator(HarmonicNumber(k-1))):k in [2..80]]; // _Marius A. Burtea_, Jul 28 2019
%o A309397 (Python)
%o A309397 from sympy import gcd, harmonic
%o A309397 def A309387(n):
%o A309397     return gcd(n**2,harmonic(n-1).p) # _Chai Wah Wu_, Jul 31 2019
%Y A309397 Cf. A001008, A007406 (see our comment), A309391.
%K A309397 nonn
%O A309397 2,2
%A A309397 _Amiram Eldar_ and _Thomas Ordowski_, Jul 28 2019