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