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A309397
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a(n) = gcd(n^2, A001008(n-1)) for n > 1.
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1
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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
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OFFSET
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2,2
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COMMENTS
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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, ...
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LINKS
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FORMULA
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a(p) = p^2 for every prime p > 3.
a(p^2) = p iff p > 3 is a prime.
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EXAMPLE
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a(11) = gcd(11^2, A001008(11-1)) = gcd(121, 7381) = 121.
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MATHEMATICA
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a[n_] := GCD[n^2, Numerator[HarmonicNumber[n-1]]]; Array[a, 72, 2]
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PROG
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(Magma) [Gcd(k^2, Numerator(HarmonicNumber(k-1))):k in [2..80]]; // Marius A. Burtea, Jul 28 2019
(Python)
from sympy import gcd, harmonic
return gcd(n**2, harmonic(n-1).p) # Chai Wah Wu, Jul 31 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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