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A126865
a(n) = gcd(Product_{p|n} (p+1)^b(p,n), Product_{p|n} (p-1)^b(p,n)), where the products are over the distinct primes, p, that divide n and p^b(p,n) is the highest power of p dividing n.
4
1, 1, 2, 1, 2, 2, 2, 1, 4, 2, 2, 2, 2, 6, 8, 1, 2, 4, 2, 2, 4, 2, 2, 2, 4, 6, 8, 6, 2, 8, 2, 1, 4, 2, 24, 4, 2, 6, 8, 2, 2, 12, 2, 2, 16, 2, 2, 2, 4, 4, 8, 6, 2, 8, 8, 6, 4, 2, 2, 8, 2, 6, 8, 1, 12, 4, 2, 2, 4, 24, 2, 4, 2, 6, 16, 18, 12, 24, 2, 2, 16, 2, 2, 12, 4, 6, 8, 2, 2, 16, 8, 2, 4, 2, 24, 2, 2, 12, 8
OFFSET
1,3
COMMENTS
First occurrence of k or 0 if not possible (including all odd primes k): 2, 1, 0, 9, 0, 14, 0, 15, 0, 0, 0, 42, 0, 0, 0, 45, 0, 76, 0, 589, 0, 0, 0, 35, 0, 0, 0, 4381, 0, 0, ..., . - Robert G. Wilson v, Sep 08 2007
FORMULA
From Antti Karttunen, Dec 17 2018: (Start)
a(n) = gcd(A003958(n), A003959(n)).
a(A007947(n)) = A066086(n).
(End)
EXAMPLE
400 = 2^4 * 5^2. So a(400) = gcd((2+1)^4 * (5+1)^2, (2-1)^4 * (5-1)^2) = gcd(2916, 16) = 4.
MATHEMATICA
f[n_] := Block[{fi = FactorInteger@n}, GCD[Times @@ ((First /@ fi - 1)^Last /@ fi), Times @@ ((First /@ fi + 1)^Last /@ fi)]]; Array[f, 99] (* Robert G. Wilson v, Sep 08 2007 *)
PROG
(PARI)
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A126865(n) = gcd(A003958(n), A003959(n)); \\ Antti Karttunen, Dec 17 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Leroy Quet, Mar 15 2007
EXTENSIONS
More terms from Robert G. Wilson v, Sep 08 2007
STATUS
approved