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Numbers whose greatest prime index is half their sum of prime indices.
20

%I #8 May 21 2021 04:18:27

%S 4,9,12,25,30,40,49,63,70,84,112,121,154,165,169,198,220,264,273,286,

%T 289,325,351,352,361,364,390,442,468,520,529,561,595,624,646,714,741,

%U 748,765,832,841,850,874,918,931,952,961,988,1020,1045,1173,1197,1224

%N Numbers whose greatest prime index is half their sum of prime indices.

%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

%F A061395(a(n)) = A056239(a(n))/2.

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

%e 4: {1,1} 198: {1,2,2,5}

%e 9: {2,2} 220: {1,1,3,5}

%e 12: {1,1,2} 264: {1,1,1,2,5}

%e 25: {3,3} 273: {2,4,6}

%e 30: {1,2,3} 286: {1,5,6}

%e 40: {1,1,1,3} 289: {7,7}

%e 49: {4,4} 325: {3,3,6}

%e 63: {2,2,4} 351: {2,2,2,6}

%e 70: {1,3,4} 352: {1,1,1,1,1,5}

%e 84: {1,1,2,4} 361: {8,8}

%e 112: {1,1,1,1,4} 364: {1,1,4,6}

%e 121: {5,5} 390: {1,2,3,6}

%e 154: {1,4,5} 442: {1,6,7}

%e 165: {2,3,5} 468: {1,1,2,2,6}

%e 169: {6,6} 520: {1,1,1,3,6}

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

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

%Y The partitions with these Heinz numbers are counted by A035363.

%Y The conjugate version is A340387.

%Y This sequence is the case of equality in A344414 and A344416.

%Y A001222 counts prime factors with multiplicity.

%Y A025065 counts palindromic partitions, ranked by A265640.

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

%Y A322109 ranks partitions of n with no part > n/2, counted by A110618.

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

%Y A344291 lists numbers m with A001222(m) <= A056239(m)/2, counted by A110618.

%Y A344296 lists numbers m with A001222(m) >= A056239(m)/2, counted by A025065.

%Y Cf. A000070, A001414, A209816, A301988, A316413, A316428, A320924, A325037, A325038, A325044, A330950, A344293, A344294, A344297.

%K nonn

%O 1,1

%A _Gus Wiseman_, May 19 2021