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Heinz numbers of integer partitions whose sum is even and is at most twice the greatest part.
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%I #9 May 21 2021 04:18:35

%S 3,4,7,9,10,12,13,19,21,22,25,28,29,30,34,37,39,40,43,46,49,52,53,55,

%T 57,61,62,63,66,70,71,76,79,82,84,85,87,88,89,91,94,101,102,107,111,

%U 112,113,115,116,117,118,121,129,130,131,133,134,136,138,139,146

%N Heinz numbers of integer partitions whose sum is even and is at most twice the greatest part.

%C 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.

%C Also numbers m whose sum of prime indices A056239(m) is even and is at most twice the greatest prime index A061395(m).

%F Intersection of A300061 and A344414.

%e The sequence of terms together with their prime indices begins:

%e 3: {2} 37: {12} 71: {20}

%e 4: {1,1} 39: {2,6} 76: {1,1,8}

%e 7: {4} 40: {1,1,1,3} 79: {22}

%e 9: {2,2} 43: {14} 82: {1,13}

%e 10: {1,3} 46: {1,9} 84: {1,1,2,4}

%e 12: {1,1,2} 49: {4,4} 85: {3,7}

%e 13: {6} 52: {1,1,6} 87: {2,10}

%e 19: {8} 53: {16} 88: {1,1,1,5}

%e 21: {2,4} 55: {3,5} 89: {24}

%e 22: {1,5} 57: {2,8} 91: {4,6}

%e 25: {3,3} 61: {18} 94: {1,15}

%e 28: {1,1,4} 62: {1,11} 101: {26}

%e 29: {10} 63: {2,2,4} 102: {1,2,7}

%e 30: {1,2,3} 66: {1,2,5} 107: {28}

%e 34: {1,7} 70: {1,3,4} 111: {2,12}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],EvenQ[Total[primeMS[#]]]&&Max[primeMS[#]]>=Total[primeMS[#]]/2&]

%Y These partitions are counted by A000070 = even-indexed terms of A025065.

%Y The opposite version appears to be A320924, counted by A209816.

%Y The opposite version with odd weights allowed appears to be A322109.

%Y The conjugate opposite version allowing odds is A344291, counted by A110618.

%Y The conjugate version is A344296, also counted by A025065.

%Y The conjugate opposite version is A344413, counted by A209816.

%Y Allowing odd weight gives A344414.

%Y The case of equality is A344415, counted by A035363.

%Y A001222 counts prime factors with multiplicity.

%Y A027187 counts partitions of even length, ranked by A028260.

%Y A056239 adds up prime indices, row sums of A112798.

%Y A058696 counts partitions of even numbers, ranked by A300061.

%Y A265640 lists Heinz numbers of palindromic partitions.

%Y A301987 lists numbers whose sum of prime indices equals their product.

%Y A334201 adds up all prime indices except the greatest.

%Y A340387 lists Heinz numbers of partitions whose sum is twice their length.

%Y Cf. A001414, A074761, A316413, A316428, A325037, A325038, A325044, A330950, A344293, A344294, A344297.

%K nonn

%O 1,1

%A _Gus Wiseman_, May 20 2021