Subject: S. M. Ulam's Lucky Numbers
1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69,
73, 75, 79, 87, 93, 99, ...
These are generated by a sieving process:
Start with all positiv integers.
A(1) = { 1,2,3,4,5,6,7,8,9,10,11,12,... }
discard every second number.
A(2) = { 1,3,5,7,9,11,13,15,17,19,21,23,25... }
the first remaning number is 3.
discard every third number.
A(3) = { 1,3,7,9,13,15,19,21,25,... }
the next remaning number is 7.
discard every 7th number.
A(4) = { 1,3,7,9,13,15,21,25,... }
the next remaning number is 9.
discard every 9th number.
. . .
A(oo) are the lucky numbers as they remain.
Ref:
Ogilvy, Mathematische Leckerbissen, Vieweg, 1980
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There is also a 'Ulam Sequence'.
a(1) = 1,
a(n+1) = a(n) + a(X(n))
with X(n) an equidistributed random variable between 1 and n.
Expected value of a(a):
E(a(n)) ~ exp(2*sqrt(n))
Ref:
M. Kac, Adv. Appl. Math. 10 (1989) 270-277