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A326584
a(n) = gcd(n*N(n-1), D(n-1)), with N(n)/D(n) = B(n) the n-th Bernoulli number.
1
1, 2, 3, 1, 5, 1, 7, 1, 3, 1, 11, 1, 13, 1, 3, 1, 17, 1, 19, 1, 3, 1, 23, 1, 5, 1, 3, 1, 29, 1, 31, 1, 3, 1, 1, 1, 37, 1, 3, 1, 41, 1, 43, 1, 15, 1, 47, 1, 7, 1, 3, 1, 53, 1, 1, 1, 3, 1, 59, 1, 61, 1, 3, 1, 5, 1, 67, 1, 3, 1, 71, 1, 73, 1, 3, 1, 1, 1, 79, 1
OFFSET
1,2
COMMENTS
Conjectures:
(1) If n > 1 then a(n) = n <=> n is prime or Carmichael (A002997).
(2) If n is odd then a(n) = 1 <=> n = 1 or is a term of A121707.
(3) The fixed points of n^2/a(n) are exactly the numbers satisfying Korselt's criterion (compare A326578 and A324050).
LINKS
FORMULA
a(n) divides n, n/a(n) = A326478(n).
EXAMPLE
a(559) = 1 and 559 is in A121707.
a(561) = 561 and 561 is Carmichael.
a(563) = 563 and 563 is prime.
MAPLE
db := n -> denom(bernoulli(n)): nb := n -> numer(bernoulli(n)):
a := n -> igcd(n*nb(n-1), db(n-1)): seq(a(n), n=1..80);
MATHEMATICA
a[n_] := With[{b = BernoulliB[n-1]}, GCD[n Numerator[b], Denominator[b]]];
Array[a, 80] (* Jean-François Alcover, Jul 21 2019 *)
PROG
(PARI) a(n) = my(b=bernfrac(n-1)); gcd(n*numerator(b), denominator(b)); \\ Michel Marcus, Jul 19 2019
CROSSREFS
Cf. A000040, A002997, A121707, A027641/A027642 (Bernoulli), A324050 (Korselt).
Sequence in context: A194446 A251758 A250480 * A166333 A322966 A173239
KEYWORD
nonn
AUTHOR
Peter Luschny, Jul 19 2019
STATUS
approved