cp's OEIS Frontend

A340605 Heinz numbers of integer partitions of even positive rank.

%I A340605 #12 Apr 09 2021 09:41:01
%S A340605 5,11,14,17,21,23,26,31,35,38,39,41,44,47,49,57,58,59,65,66,67,68,73,
%T A340605 74,83,86,87,91,92,95,97,99,102,103,104,106,109,110,111,122,124,127,
%U A340605 129,133,137,138,142,143,145,149,152,153,154,156,157,158,159,164,165
%N A340605 Heinz numbers of integer partitions of even positive rank.
%C A340605 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 A340605 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 A340605 FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000145">St000145: The Dyson rank of a partition</a>
%F A340605 A061395(a(n)) - A001222(a(n)) is even and positive.
%F A340605 A340604 \/ A340605 = A340787.
%e A340605 The sequence of partitions with their Heinz numbers begins:
%e A340605       5: (3)         57: (8,2)       97: (25)
%e A340605      11: (5)         58: (10,1)      99: (5,2,2)
%e A340605      14: (4,1)       59: (17)       102: (7,2,1)
%e A340605      17: (7)         65: (6,3)      103: (27)
%e A340605      21: (4,2)       66: (5,2,1)    104: (6,1,1,1)
%e A340605      23: (9)         67: (19)       106: (16,1)
%e A340605      26: (6,1)       68: (7,1,1)    109: (29)
%e A340605      31: (11)        73: (21)       110: (5,3,1)
%e A340605      35: (4,3)       74: (12,1)     111: (12,2)
%e A340605      38: (8,1)       83: (23)       122: (18,1)
%e A340605      39: (6,2)       86: (14,1)     124: (11,1,1)
%e A340605      41: (13)        87: (10,2)     127: (31)
%e A340605      44: (5,1,1)     91: (6,4)      129: (14,2)
%e A340605      47: (15)        92: (9,1,1)    133: (8,4)
%e A340605      49: (4,4)       95: (8,3)      137: (33)
%t A340605 rk[n_]:=PrimePi[FactorInteger[n][[-1,1]]]-PrimeOmega[n];
%t A340605 Select[Range[100],EvenQ[rk[#]]&&rk[#]>0&]
%Y A340605 Note: Heinz numbers are given in parentheses below.
%Y A340605 Allowing any positive rank gives A064173 (A340787).
%Y A340605 The odd version is counted by A101707 (A340604).
%Y A340605 These partitions are counted by A101708.
%Y A340605 The not necessarily positive case is counted by A340601 (A340602).
%Y A340605 A001222 counts prime indices.
%Y A340605 A061395 gives maximum prime index.
%Y A340605 A072233 counts partitions by sum and length.
%Y A340605 - Rank -
%Y A340605 A047993 counts partitions of rank 0 (A106529).
%Y A340605 A064173 counts partitions of negative rank (A340788).
%Y A340605 A064174 counts partitions of nonnegative rank (A324562).
%Y A340605 A064174 (also) counts partitions of nonpositive rank (A324521).
%Y A340605 A101198 counts partitions of rank 1 (A325233).
%Y A340605 A257541 gives the rank of the partition with Heinz number n.
%Y A340605 A340692 counts partitions of odd rank (A340603).
%Y A340605 - Even -
%Y A340605 A027187 counts partitions of even length (A028260).
%Y A340605 A027187 (also) counts partitions of even maximum (A244990).
%Y A340605 A035363 counts partitions into even parts (A066207).
%Y A340605 A058696 counts partitions of even numbers (A300061).
%Y A340605 A067661 counts strict partitions of even length (A030229).
%Y A340605 A339846 counts factorizations of even length.
%Y A340605 Cf. A006141, A024430, A056239, A112798, A340387, A340598, A340600, A340608, A340609, A340610, A340653.
%K A340605 nonn
%O A340605 1,1
%A A340605 _Gus Wiseman_, Jan 21 2021