cp's OEIS Frontend

A340611 Number of integer partitions of n of length 2^k where k is the greatest part.

%I A340611 #7 Jan 30 2021 22:50:56
%S A340611 1,0,1,0,0,1,1,1,1,0,1,1,2,2,3,3,4,4,4,4,4,4,5,5,6,7,8,10,12,14,16,19,
%T A340611 21,24,27,29,32,34,36,38,41,42,45,47,50,52,56,58,63,66,71,75,83,88,98,
%U A340611 106,118,128,143,155,173,188,208,226,250,270,297,321,350
%N A340611 Number of integer partitions of n of length 2^k where k is the greatest part.
%C A340611 Also the number of integer partitions of n with maximum 2^k where k is the length.
%e A340611 The partitions for n = 12, 14, 16, 22, 24:
%e A340611   32211111  32222111  32222221  33333322          33333333
%e A340611   33111111  33221111  33222211  33333331          4222221111111111
%e A340611             33311111  33322111  4222111111111111  4322211111111111
%e A340611                       33331111  4321111111111111  4332111111111111
%e A340611                                 4411111111111111  4422111111111111
%e A340611                                                   4431111111111111
%e A340611 The conjugate partitions:
%e A340611   (8,2,2)  (8,3,3)  (8,4,4)  (8,7,7)     (8,8,8)
%e A340611   (8,3,1)  (8,4,2)  (8,5,3)  (8,8,6)     (16,3,3,2)
%e A340611            (8,5,1)  (8,6,2)  (16,2,2,2)  (16,4,2,2)
%e A340611                     (8,7,1)  (16,3,2,1)  (16,4,3,1)
%e A340611                              (16,4,1,1)  (16,5,2,1)
%e A340611                                          (16,6,1,1)
%t A340611 Table[Length[Select[IntegerPartitions[n],Length[#]==2^Max@@#&]],{n,0,30}]
%Y A340611 Note: A-numbers of Heinz-number sequences are in parentheses below.
%Y A340611 A018818 counts partitions of n into divisors of n (A326841).
%Y A340611 A047993 counts balanced partitions (A106529).
%Y A340611 A067538 counts partitions of n whose length/max divides n (A316413/A326836).
%Y A340611 A072233 counts partitions by sum and length.
%Y A340611 A168659 = partitions whose greatest part divides their length (A340609).
%Y A340611 A168659 = partitions whose length divides their greatest part (A340610).
%Y A340611 A326843 = partitions of n whose length and maximum both divide n (A326837).
%Y A340611 A340597 lists numbers with an alt-balanced factorization.
%Y A340611 A340653 counts balanced factorizations.
%Y A340611 A340689 have a factorization of length 2^max.
%Y A340611 A340690 have a factorization of maximum 2^length.
%Y A340611 Cf. A003114, A006141, A064174, A098124, A117409, A200750, A340385, A340387, A340599, A340601.
%K A340611 nonn
%O A340611 0,13
%A A340611 _Gus Wiseman_, Jan 28 2021