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A309391
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a(n) = gcd(n, A064169(n-2)) for n > 2.
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2
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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
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OFFSET
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3,1
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COMMENTS
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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, ...
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LINKS
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FORMULA
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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.
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EXAMPLE
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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, ...
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MAPLE
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H:= 0:
for n from 3 to 100 do
H:= H + 1/(n-2);
A[n]:= igcd(n, numer(H)-denom(H));
od:
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MATHEMATICA
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a[n_] := GCD[n, Numerator[(h = HarmonicNumber[n-2])] - Denominator[h]]; Array[a, 81, 3]
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PROG
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(Magma) [Gcd(k, Numerator(a)-Denominator(a)) where a is HarmonicNumber(k-2):k in [3..90]]; // Marius A. Burtea, Jul 29 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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