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A180149
Integers with precisely two partitions into sums of four squares of nonnegative numbers.
12
4, 9, 10, 12, 13, 16, 17, 19, 20, 21, 22, 29, 30, 31, 35, 39, 40, 44, 46, 47, 48, 64, 71, 80, 88, 120, 160, 176, 184, 192, 256, 320, 352, 480, 640, 704, 736, 768, 1024, 1280, 1408, 1920, 2560, 2816, 2944, 3072, 4096, 5120, 5632, 7680
OFFSET
1,1
COMMENTS
The largest odd member of this sequence is 71, and from a(32)=320 onwards the terms satisfy the eighth-order recurrence relation a(n)=4a(n-8).
A002635(a(n)) = 2. - Reinhard Zumkeller, Jul 13 2014
LINKS
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No.8, October 1948, pp. 476-481.
FORMULA
The members of this sequence are {9, 13, 17, 19, 21, 29, 30, 31, 35, 39, 46, 47, 71} together with all integers of the form 5*2^N, 11*2^N and {1,3,30,46}*4^N where N > 0 (which includes a necessary correction to Lehmer's result).
EXAMPLE
As the fifth integer which has precisely two partitions into sums of four squares of nonnegative numbers is 13, then a(5)=13.
MATHEMATICA
Select[Range[10000], Length[PowersRepresentations[ #, 4, 2]]==2&]
PROG
(Haskell)
a180149 n = a180149_list !! (n-1)
a180149_list = filter ((== 2) . a002635) [0..]
-- Reinhard Zumkeller, Jul 13 2014
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Ant King, Aug 17 2010
STATUS
approved