A359903 - OEIS

%I #7 Jan 26 2023 10:04:23

%S 1,2,9,88,100,125,624,756,792,810,880,900,1312,2401,4617,4624,6240,

%T 7392,7560,7920,8400,9261,9604,9801,10648,12416,23424,33984,37760,

%U 45792,47488,60912,66176,71552,73920,75200,78720,83592,89216,89984,91264,91648,99456

%N Numbers whose prime indices and prime signature have the same 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.

%e The terms together with their prime indices begin:

%e       1: {}

%e       2: {1}

%e       9: {2,2}

%e      88: {1,1,1,5}

%e     100: {1,1,3,3}

%e     125: {3,3,3}

%e     624: {1,1,1,1,2,6}

%e     756: {1,1,2,2,2,4}

%e     792: {1,1,1,2,2,5}

%e     810: {1,2,2,2,2,3}

%e     880: {1,1,1,1,3,5}

%e     900: {1,1,2,2,3,3}

%e    1312: {1,1,1,1,1,13}

%e    2401: {4,4,4,4}

%e    4617: {2,2,2,2,2,8}

%e    4624: {1,1,1,1,7,7}

%e    6240: {1,1,1,1,1,2,3,6}

%e    7392: {1,1,1,1,1,2,4,5}

%e    7560: {1,1,1,2,2,2,3,4}

%e    7920: {1,1,1,1,2,2,3,5}

%e Example: 810 has prime indices {1,2,2,2,2,3} and prime exponents (1,4,1), both of which have mean 2, so 810 is in the sequence.

%e Example: 78720 has prime indices {1,1,1,1,1,1,1,2,3,13} and prime exponents (7,1,1,1), both of which have mean 5/2, so 78720 is in the sequence.

%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[1000],Mean[prix[#]]==Mean[prisig[#]]&]

%Y Prime indices are A112798, sum A056239, mean A326567/A326568.

%Y Prime signature is A124010, sum A001222, mean A088529/A088530.

%Y For prime factors instead of indices we have A359904.

%Y Partitions with these Heinz numbers are counted by A360068.

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

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

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

%Y A360005 gives median of prime indices (times two).

%Y Cf. A240219, A316313, A326622, A327473, A327476, A348551, A359905, A359908, A360008, A360069.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 24 2023