cp's OEIS Frontend

A340604 Heinz numbers of integer partitions of odd positive rank.

%I A340604 #13 Jan 22 2021 20:29:02
%S A340604 3,7,10,13,15,19,22,25,28,29,33,34,37,42,43,46,51,52,53,55,61,62,63,
%T A340604 69,70,71,76,77,78,79,82,85,88,89,93,94,98,101,105,107,113,114,115,
%U A340604 116,117,118,119,121,123,130,131,132,134,136,139,141,146,147,148,151
%N A340604 Heinz numbers of integer partitions of odd positive rank.
%C A340604 The Dyson rank of a nonempty partition is its maximum part minus its number of parts. The rank of an empty partition is 0.
%C A340604 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%H A340604 FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000145">St000145: The Dyson rank of a partition</a>
%F A340604 A061395(a(n)) - A001222(a(n)) is odd and positive.
%F A340604 A340604 \/ A340605 = A340787.
%e A340604 The sequence of partitions with their Heinz numbers begins:
%e A340604       3: (2)         46: (9,1)       82: (13,1)
%e A340604       7: (4)         51: (7,2)       85: (7,3)
%e A340604      10: (3,1)       52: (6,1,1)     88: (5,1,1,1)
%e A340604      13: (6)         53: (16)        89: (24)
%e A340604      15: (3,2)       55: (5,3)       93: (11,2)
%e A340604      19: (8)         61: (18)        94: (15,1)
%e A340604      22: (5,1)       62: (11,1)      98: (4,4,1)
%e A340604      25: (3,3)       63: (4,2,2)    101: (26)
%e A340604      28: (4,1,1)     69: (9,2)      105: (4,3,2)
%e A340604      29: (10)        70: (4,3,1)    107: (28)
%e A340604      33: (5,2)       71: (20)       113: (30)
%e A340604      34: (7,1)       76: (8,1,1)    114: (8,2,1)
%e A340604      37: (12)        77: (5,4)      115: (9,3)
%e A340604      42: (4,2,1)     78: (6,2,1)    116: (10,1,1)
%e A340604      43: (14)        79: (22)       117: (6,2,2)
%t A340604 rk[n_]:=PrimePi[FactorInteger[n][[-1,1]]]-PrimeOmega[n];
%t A340604 Select[Range[100],OddQ[rk[#]]&&rk[#]>0&]
%Y A340604 Note: Heinz numbers are given in parentheses below.
%Y A340604 These partitions are counted by A101707.
%Y A340604 Allowing negative ranks gives A340692, counted by A340603.
%Y A340604 The even version is A340605, counted by A101708.
%Y A340604 The not necessarily odd case is A340787, counted by A064173.
%Y A340604 A001222 gives number of prime indices.
%Y A340604 A061395 gives maximum prime index.
%Y A340604 - Rank -
%Y A340604 A047993 counts partitions of rank 0 (A106529).
%Y A340604 A064173 counts partitions of negative rank (A340788).
%Y A340604 A064174 counts partitions of nonnegative rank (A324562).
%Y A340604 A064174 (also) counts partitions of nonpositive rank (A324521).
%Y A340604 A101198 counts partitions of rank 1 (A325233).
%Y A340604 A257541 gives the rank of the partition with Heinz number n.
%Y A340604 A340653 counts balanced factorizations.
%Y A340604 - Odd -
%Y A340604 A000009 counts partitions into odd parts (A066208).
%Y A340604 A027193 counts partitions of odd length (A026424).
%Y A340604 A027193 (also) counts partitions of odd maximum (A244991).
%Y A340604 A058695 counts partitions of odd numbers (A300063).
%Y A340604 A067659 counts strict partitions of odd length (A030059).
%Y A340604 A160786 counts odd-length partitions of odd numbers (A300272).
%Y A340604 A339890 counts factorizations of odd length.
%Y A340604 A340101 counts factorizations into odd factors.
%Y A340604 A340102 counts odd-length factorizations into odd factors.
%Y A340604 A340385 counts partitions of odd length and maximum (A340386).
%Y A340604 Cf. A001221, A006141, A056239, A112798, A168659, A200750, A316413, A325134, A340601, A340602, A340608, A340609, A340610.
%K A340604 nonn
%O A340604 1,1
%A A340604 _Gus Wiseman_, Jan 21 2021