A034268 a(n) = LCM_{k=1..n} (2^k - 1).
1, 3, 21, 105, 3255, 9765, 1240155, 21082635, 1539032355, 16929355905, 34654391537535, 450507089987955, 3690103574091339405, 158674453685927594415, 23959842506575066756665, 6157679524189792156462905, 807093212915080247739749421255
Offset: 1
Keywords
Examples
a(3) = lcm(1,3,7) = 21.
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..100
- Gert Almkvist, Powers of a matrix with coefficients in a Boolean ring, Proc. Amer. Math. Soc. 53 (1975), 27-31. See v_n.
- J. B. Marshall, On the extension of Fermat's theorem to matrices of order n, Proceedings of the Edinburgh Mathematical Society 6 (1939) 85-91. See (10) page 90 for p=2.
- MathOverflow, Do we know any bound on lcm(2^1-1, 2^2-1, ..., 2^n-1)?
Programs
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Magma
[Lcm([2^k-1:k in [1..n]]): n in [1..17]]; // Marius A. Burtea, Jan 29 2020
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Maple
a:= proc(n) option remember; `if`(n=1, 1, ilcm(a(n-1), 2^n-1)) end: seq(a(n), n=1..20); # Alois P. Heinz, Oct 16 2011
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Mathematica
Table[LCM @@ (2^Range[n] - 1), {n, 1, 20}] (* Jean-François Alcover, Apr 02 2015 *) -
PARI
A034268(n) = {local(r);r=1;for(k=1,n,r=lcm(r,2^k-1));r} \\ Michael B. Porter, Mar 02 2010
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PARI
a(n) = lcm(vector(n, k, 2^k-1)); \\ Michel Marcus, Jul 29 2022
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Python
from math import lcm from itertools import accumulate def aupto(n): return list(accumulate((2**k-1 for k in range(1, n+1)), lcm)) print(aupto(17)) # Michael S. Branicky, Jul 04 2022
Formula
a(n) = lcm(1, 3, 7, ..., 2^n - 1).
a(n) = Product_{k=1..n} Phi_k(2), where Phi_n(2) is n-th cyclotomic polynomial at x=2 (cf. A019320). - Vladeta Jovovic, Jan 20 2002