cp's OEIS Frontend

A372688 Number of integer partitions y of n whose rank Sum_i 2^(y_i-1) is prime.

%I A372688 #6 May 17 2024 19:47:26
%S A372688 0,0,2,2,1,3,3,6,3,6,9,20,13,22,22,45,47,70,75,100,107,132,157,202,
%T A372688 229,302,396,495,536,699,820,962,1193,1507,1699,2064,2455,2945,3408,
%U A372688 4026,4691,5749,6670,7614,9127,10930,12329,14370,16955,19961,22950,26574,30941
%N A372688 Number of integer partitions y of n whose rank Sum_i 2^(y_i-1) is prime.
%C A372688 Note the function taking a set s to Sum_i 2^(s_i-1) is the inverse of A048793 (binary indices).
%e A372688 The partition (3,2,1) has rank 2^(3-1) + 2^(2-1) + 2^(1-1) = 7, which is prime, so (3,2,1) is counted under a(6).
%e A372688 The a(2) = 2 through a(10) = 9 partitions:
%e A372688 (2)   (21)   (31)  (221)    (51)    (421)      (431)   (441)     (91)
%e A372688 (11)  (111)        (2111)   (321)   (2221)     (521)   (3321)    (631)
%e A372688                    (11111)  (3111)  (4111)     (5111)  (4221)    (721)
%e A372688                                     (22111)            (33111)   (3331)
%e A372688                                     (211111)           (42111)   (7111)
%e A372688                                     (1111111)          (411111)  (32221)
%e A372688                                                                  (322111)
%e A372688                                                                  (3211111)
%e A372688                                                                  (31111111)
%t A372688 Table[Length[Select[IntegerPartitions[n], PrimeQ[Total[2^#]/2]&]],{n,0,30}]
%Y A372688 For all positive integers (not just prime) we get A000041.
%Y A372688 For even instead of prime we have A087787, strict A025147, odd A096765.
%Y A372688 These partitions have Heinz numbers A277319.
%Y A372688 The strict case is A372687, ranks A372851.
%Y A372688 The version counting only distinct parts is A372887, ranks A372850.
%Y A372688 A014499 lists binary indices of prime numbers.
%Y A372688 A019565 gives Heinz number of binary indices, adjoint A048675.
%Y A372688 A048793 and A272020 (reverse) list binary indices:
%Y A372688 - length A000120
%Y A372688 - min A001511
%Y A372688 - sum A029931
%Y A372688 - max A070939
%Y A372688 A058698 counts partitions of prime numbers, strict A064688.
%Y A372688 A372885 lists primes whose binary indices sum to a prime, indices A372886.
%Y A372688 Cf. A000040, A005940, A023506, A029837, A035100, A038499, A096111, A372429, A372441, A372471, A372689.
%K A372688 nonn
%O A372688 0,3
%A A372688 _Gus Wiseman_, May 16 2024