

A051179


a(n) = 2^(2^n)  1.


40



1, 3, 15, 255, 65535, 4294967295, 18446744073709551615, 340282366920938463463374607431768211455, 115792089237316195423570985008687907853269984665640564039457584007913129639935
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OFFSET

0,2


COMMENTS

In a tree with binary nodes (0, 1 children only), the maximum number of unique child nodes at level n.
Number of binary trees (each vertex has 0, or 1 left, or 1 right, or 2 children) such that all leaves are at level n. Example: a(1) = 3 because we have (i) root with a left child, (ii) root with a right child and (iii) root with two children. a(n) = A000215(n)  2.  Emeric Deutsch, Jan 20 2004
Similarly, this is also the number of full balanced binary trees of height n. (There is an obvious 1to1 correspondence between the two sets of trees.)  David Hobby (hobbyd(AT)newpaltz.edu), May 02 2010
The first 5 terms n (only) have the property that phi(n)=(n+1)/2, where phi(n) = A000010(n) is Euler's totient function.  Lekraj Beedassy, Feb 12 2007
For any odd positive a(0), the sequence defined by a(n) = a(n1) * (a(n1) + 2) gives a constructive proof that there exist integers with at least n distinct prime factors, e.g., a(n), since omega(a(n)) >= n. As a corollary, this gives a constructive proof of Euclid's theorem stating that there are infinitely many primes.  Daniel Forgues, Mar 07 2017
I conjecture that, for n > 7, omega(a(n)) > omega(a(n1)) > n.
It seems that the largest prime divisor p(n+1) of a(n+1) is always bigger than the largest prime divisor of a(n): p(n+1) > p(n). For 3 < n < 8, p(n+1) > 100 * p(n).
(End)


REFERENCES

M. Aigner and G. M. Ziegler, Proofs from The Book, SpringerVerlag, Berlin, 1999; see p. 4.
Ben Delo and Filip Saidak, Euclid's theorem redux, Fib. Q., 57:4 (2019), 331336.


LINKS



FORMULA

a(n) = (a(n1) + 1)^2  1, a(0) = 1. [ or a(n) = a(n1)(a(n1) + 2) ].
1 = 2/3 + 4/15 + 16/255 + 256/65535 + ... = Sum_{n>=0} A001146(n)/a(n+1) with partial sums: 2/3, 14/15, 254/255, 65534/65535, ...  Gary W. Adamson, Jun 15 2003
a(n) = b(n1) where b(1)=1, b(n) = Product_{k=1..n1} (b(k) + 2).  Benoit Cloitre, Sep 13 2003


EXAMPLE

15 = 3*5; 255 = 3*5*17; 65535 = 3*5*17*257; ...  Daniel Forgues, Mar 07 2017


MAPLE



MATHEMATICA



PROG

(PARI) a(n)=if(n<0, 0, 2^2^n1)
(Python)


CROSSREFS



KEYWORD

nonn,easy,nice


AUTHOR

Alan DeKok (aland(AT)ox.org)


STATUS

approved



