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A344414 Heinz numbers of integer partitions whose sum is at most twice their greatest part.

%I A344414 #12 May 21 2021 04:18:16
%S A344414 2,3,4,5,6,7,9,10,11,12,13,14,15,17,19,20,21,22,23,25,26,28,29,30,31,
%T A344414 33,34,35,37,38,39,40,41,42,43,44,46,47,49,51,52,53,55,56,57,58,59,61,
%U A344414 62,63,65,66,67,68,69,70,71,73,74,76,77,78,79,82,83,84,85
%N A344414 Heinz numbers of integer partitions whose sum is at most twice their greatest part.
%C A344414 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
%F A344414 A056239(a(n)) <= 2*A061395(a(n)).
%e A344414 The sequence of terms together with their prime indices begins:
%e A344414      2: {1}        20: {1,1,3}    39: {2,6}
%e A344414      3: {2}        21: {2,4}      40: {1,1,1,3}
%e A344414      4: {1,1}      22: {1,5}      41: {13}
%e A344414      5: {3}        23: {9}        42: {1,2,4}
%e A344414      6: {1,2}      25: {3,3}      43: {14}
%e A344414      7: {4}        26: {1,6}      44: {1,1,5}
%e A344414      9: {2,2}      28: {1,1,4}    46: {1,9}
%e A344414     10: {1,3}      29: {10}       47: {15}
%e A344414     11: {5}        30: {1,2,3}    49: {4,4}
%e A344414     12: {1,1,2}    31: {11}       51: {2,7}
%e A344414     13: {6}        33: {2,5}      52: {1,1,6}
%e A344414     14: {1,4}      34: {1,7}      53: {16}
%e A344414     15: {2,3}      35: {3,4}      55: {3,5}
%e A344414     17: {7}        37: {12}       56: {1,1,1,4}
%e A344414     19: {8}        38: {1,8}      57: {2,8}
%e A344414 For example, 56 has prime indices {1,1,1,4} and 7 <= 2*4, so 56 is in the sequence. On the other hand, 224 has prime indices {1,1,1,1,1,4} and 9 > 2*4, so 224 is not in the sequence.
%t A344414 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A344414 Select[Range[100],Max[primeMS[#]]>=Total[primeMS[#]]/2&]
%Y A344414 These partitions are counted by A025065 but are different from palindromic partitions, which have Heinz numbers A265640.
%Y A344414 The opposite even-weight version appears to be A320924, counted by A209816.
%Y A344414 The opposite version appears to be A322109, counted by A110618.
%Y A344414 The case of equality in the conjugate version is A340387.
%Y A344414 The conjugate opposite version is A344291, counted by A110618.
%Y A344414 The conjugate opposite 5-smooth case is A344293, counted by A266755.
%Y A344414 The conjugate version is A344296, also counted by A025065.
%Y A344414 The case of equality is A344415.
%Y A344414 The even-weight case is A344416.
%Y A344414 A001222 counts prime factors with multiplicity.
%Y A344414 A027187 counts partitions of even length, ranked by A028260.
%Y A344414 A056239 adds up prime indices, row sums of A112798.
%Y A344414 A058696 counts partitions of even numbers, ranked by A300061.
%Y A344414 A301987 lists numbers whose sum of prime indices equals their product.
%Y A344414 A334201 adds up all prime indices except the greatest.
%Y A344414 Cf. A001414, A067538, A301988, A316413, A316428, A325037, A325038, A325044, A330950, A344294, A344297.
%K A344414 nonn
%O A344414 1,1
%A A344414 _Gus Wiseman_, May 19 2021