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Numbers whose prime indices and prime signature both have integer mean.
12

%I #6 Jan 26 2023 10:04:28

%S 2,3,4,5,7,8,9,10,11,13,16,17,19,21,22,23,25,27,29,30,31,32,34,37,39,

%T 41,43,46,47,49,53,55,57,59,61,62,64,67,71,73,78,79,81,82,83,85,87,88,

%U 89,91,94,97,100,101,103,105,107,109,110,111,113,115,118,121

%N Numbers whose prime indices and prime signature both have integer mean.

%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.

%C A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

%F Intersection of A316413 and A067340.

%e The terms together with their prime indices begin:

%e 2: {1} 19: {8} 39: {2,6}

%e 3: {2} 21: {2,4} 41: {13}

%e 4: {1,1} 22: {1,5} 43: {14}

%e 5: {3} 23: {9} 46: {1,9}

%e 7: {4} 25: {3,3} 47: {15}

%e 8: {1,1,1} 27: {2,2,2} 49: {4,4}

%e 9: {2,2} 29: {10} 53: {16}

%e 10: {1,3} 30: {1,2,3} 55: {3,5}

%e 11: {5} 31: {11} 57: {2,8}

%e 13: {6} 32: {1,1,1,1,1} 59: {17}

%e 16: {1,1,1,1} 34: {1,7} 61: {18}

%e 17: {7} 37: {12} 62: {1,11}

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

%t prisig[n_]:=If[n==1,{},Last/@FactorInteger[n]];

%t Select[Range[100],IntegerQ[Mean[prix[#]]]&&IntegerQ[Mean[prisig[#]]]&]

%Y A058398 counts partitions by mean, see also A008284, A327482.

%Y A067340 lists numbers whose prime signature has integer mean.

%Y A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.

%Y A124010 lists prime signature, mean A088529/A088530.

%Y A316413 lists numbers whose prime indices have integer mean.

%Y A326622 counts factorizations with integer mean, strict A328966.

%Y Cf. A327473, A348551, A359904, A359908, A360005, A360068, A360069.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jan 25 2023