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A348551 Heinz numbers of integer partitions whose mean is not an integer.

%I A348551 #13 Nov 20 2021 23:47:44
%S A348551 1,6,12,14,15,18,20,24,26,33,35,36,38,40,42,44,45,48,50,51,52,54,56,
%T A348551 58,60,63,65,66,69,70,72,74,75,76,77,80,86,92,93,95,96,102,104,106,
%U A348551 108,112,114,117,119,120,122,123,124,126,130,132,135,136,140,141,142
%N A348551 Heinz numbers of integer partitions whose mean is not an integer.
%C A348551 Equivalently, partitions whose length does not divide their sum.
%C A348551 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.
%e A348551 The terms and their prime indices begin:
%e A348551    1: {}
%e A348551    6: {1,2}
%e A348551   12: {1,1,2}
%e A348551   14: {1,4}
%e A348551   15: {2,3}
%e A348551   18: {1,2,2}
%e A348551   20: {1,1,3}
%e A348551   24: {1,1,1,2}
%e A348551   26: {1,6}
%e A348551   33: {2,5}
%e A348551   35: {3,4}
%e A348551   36: {1,1,2,2}
%e A348551   38: {1,8}
%e A348551   40: {1,1,1,3}
%e A348551   42: {1,2,4}
%e A348551   44: {1,1,5}
%e A348551   45: {2,2,3}
%e A348551   48: {1,1,1,1,2}
%p A348551 q:= n-> (l-> nops(l)=0 or irem(add(i, i=l), nops(l))>0)(map
%p A348551         (i-> numtheory[pi](i[1])$i[2], ifactors(n)[2])):
%p A348551 select(q, [$1..142])[];  # _Alois P. Heinz_, Nov 19 2021
%t A348551 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A348551 Select[Range[100],!IntegerQ[Mean[primeMS[#]]]&]
%Y A348551 A version counting nonempty subsets is A000079 - A051293.
%Y A348551 A version counting factorizations is A001055 - A326622.
%Y A348551 A version counting compositions is A011782 - A271654.
%Y A348551 A version for prime factors is A175352, complement A078175.
%Y A348551 A version for distinct prime factors A176587, complement A078174.
%Y A348551 The complement is A316413, counted by A067538, strict A102627.
%Y A348551 The geometric version is the complement of A326623.
%Y A348551 The conjugate version is the complement of A326836.
%Y A348551 These partitions are counted by A349156.
%Y A348551 A000041 counts partitions.
%Y A348551 A001222 counts prime factors with multiplicity.
%Y A348551 A018818 counts partitions into divisors, ranked by A326841.
%Y A348551 A143773 counts partitions into multiples of the length, ranked by A316428.
%Y A348551 A236634 counts unbalanced partitions.
%Y A348551 A047993 counts balanced partitions, ranked by A106529.
%Y A348551 A056239 adds up prime indices, row sums of A112798.
%Y A348551 A326567/A326568 gives the mean of prime indices, conjugate A326839/A326840.
%Y A348551 A327472 counts partitions not containing their mean, complement A237984.
%Y A348551 Cf. A067539, A096199, A098743, A175397, A175761, A289508, A289509, A290103, A326028, A326645, A326837.
%K A348551 nonn
%O A348551 1,2
%A A348551 _Gus Wiseman_, Nov 14 2021