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A344416 Heinz numbers of integer partitions whose sum is even and is at most twice the greatest part.

%I A344416 #9 May 21 2021 04:18:35
%S A344416 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 A344416 57,61,62,63,66,70,71,76,79,82,84,85,87,88,89,91,94,101,102,107,111,
%U A344416 112,113,115,116,117,118,121,129,130,131,133,134,136,138,139,146
%N A344416 Heinz numbers of integer partitions whose sum is even and is at most twice the greatest part.
%C A344416 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 A344416 Also numbers m whose sum of prime indices A056239(m) is even and is at most twice the greatest prime index A061395(m).
%F A344416 Intersection of A300061 and A344414.
%e A344416 The sequence of terms together with their prime indices begins:
%e A344416       3: {2}         37: {12}          71: {20}
%e A344416       4: {1,1}       39: {2,6}         76: {1,1,8}
%e A344416       7: {4}         40: {1,1,1,3}     79: {22}
%e A344416       9: {2,2}       43: {14}          82: {1,13}
%e A344416      10: {1,3}       46: {1,9}         84: {1,1,2,4}
%e A344416      12: {1,1,2}     49: {4,4}         85: {3,7}
%e A344416      13: {6}         52: {1,1,6}       87: {2,10}
%e A344416      19: {8}         53: {16}          88: {1,1,1,5}
%e A344416      21: {2,4}       55: {3,5}         89: {24}
%e A344416      22: {1,5}       57: {2,8}         91: {4,6}
%e A344416      25: {3,3}       61: {18}          94: {1,15}
%e A344416      28: {1,1,4}     62: {1,11}       101: {26}
%e A344416      29: {10}        63: {2,2,4}      102: {1,2,7}
%e A344416      30: {1,2,3}     66: {1,2,5}      107: {28}
%e A344416      34: {1,7}       70: {1,3,4}      111: {2,12}
%t A344416 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A344416 Select[Range[100],EvenQ[Total[primeMS[#]]]&&Max[primeMS[#]]>=Total[primeMS[#]]/2&]
%Y A344416 These partitions are counted by A000070 = even-indexed terms of A025065.
%Y A344416 The opposite version appears to be A320924, counted by A209816.
%Y A344416 The opposite version with odd weights allowed appears to be A322109.
%Y A344416 The conjugate opposite version allowing odds is A344291, counted by A110618.
%Y A344416 The conjugate version is A344296, also counted by A025065.
%Y A344416 The conjugate opposite version is A344413, counted by A209816.
%Y A344416 Allowing odd weight gives A344414.
%Y A344416 The case of equality is A344415, counted by A035363.
%Y A344416 A001222 counts prime factors with multiplicity.
%Y A344416 A027187 counts partitions of even length, ranked by A028260.
%Y A344416 A056239 adds up prime indices, row sums of A112798.
%Y A344416 A058696 counts partitions of even numbers, ranked by A300061.
%Y A344416 A265640 lists Heinz numbers of palindromic partitions.
%Y A344416 A301987 lists numbers whose sum of prime indices equals their product.
%Y A344416 A334201 adds up all prime indices except the greatest.
%Y A344416 A340387 lists Heinz numbers of partitions whose sum is twice their length.
%Y A344416 Cf. A001414, A074761, A316413, A316428, A325037, A325038, A325044, A330950, A344293, A344294, A344297.
%K A344416 nonn
%O A344416 1,1
%A A344416 _Gus Wiseman_, May 20 2021