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A366318 Heinz numbers of integer partitions that are of length 2 or begin with n/2, where n is the sum of all parts.

%I A366318 #8 Oct 10 2023 09:30:07
%S A366318 4,6,9,10,12,14,15,21,22,25,26,30,33,34,35,38,39,40,46,49,51,55,57,58,
%T A366318 62,63,65,69,70,74,77,82,84,85,86,87,91,93,94,95,106,111,112,115,118,
%U A366318 119,121,122,123,129,133,134,141,142,143,145,146,154,155,158,159
%N A366318 Heinz numbers of integer partitions that are of length 2 or begin with n/2, where n is the sum of all parts.
%C A366318 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.
%F A366318 Union of A001358 and A344415.
%e A366318 The terms together with their prime indices begin:
%e A366318      4: {1,1}      38: {1,8}         77: {4,5}
%e A366318      6: {1,2}      39: {2,6}         82: {1,13}
%e A366318      9: {2,2}      40: {1,1,1,3}     84: {1,1,2,4}
%e A366318     10: {1,3}      46: {1,9}         85: {3,7}
%e A366318     12: {1,1,2}    49: {4,4}         86: {1,14}
%e A366318     14: {1,4}      51: {2,7}         87: {2,10}
%e A366318     15: {2,3}      55: {3,5}         91: {4,6}
%e A366318     21: {2,4}      57: {2,8}         93: {2,11}
%e A366318     22: {1,5}      58: {1,10}        94: {1,15}
%e A366318     25: {3,3}      62: {1,11}        95: {3,8}
%e A366318     26: {1,6}      63: {2,2,4}      106: {1,16}
%e A366318     30: {1,2,3}    65: {3,6}        111: {2,12}
%e A366318     33: {2,5}      69: {2,9}        112: {1,1,1,1,4}
%e A366318     34: {1,7}      70: {1,3,4}      115: {3,9}
%e A366318     35: {3,4}      74: {1,12}       118: {1,17}
%t A366318 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A366318 Select[Range[100],Length[prix[#]]==2||MemberQ[prix[#],Total[prix[#]]/2]&]
%Y A366318 The first condition alone is A001358, counted by A004526.
%Y A366318 The complement of the first condition is A100959, counted by A058984.
%Y A366318 The partitions with these Heinz numbers are counted by A238628.
%Y A366318 The second condition alone is A344415, counted by A035363.
%Y A366318 The complement of the second condition is A366319, counted by A086543.
%Y A366318 A001222 counts prime factors with multiplicity.
%Y A366318 A056239 adds up prime indices, row sums of A112798.
%Y A366318 A322109 ranks partitions of n with no part > n/2, counted by A110618.
%Y A366318 A334201 adds up all prime indices except the greatest.
%Y A366318 A344296 solves for k in A001222(k) >= A056239(k)/2, counted by A025065.
%Y A366318 Cf. A001414, A316413, A320924, A325044, A340387, A344414, A344416.
%K A366318 nonn
%O A366318 1,1
%A A366318 _Gus Wiseman_, Oct 08 2023