A363951 - OEIS

%I #7 Jul 06 2023 08:55:16

%S 2,9,10,68,78,98,99,105,110,125,328,444,558,620,783,812,870,966,988,

%T 1012,1035,1150,1156,1168,1197,1254,1326,1330,1425,1521,1666,1683,

%U 1690,1704,1785,1870,1911,2002,2125,2145,2275,2401,2412,2541,2662,2680,2695,3025

%N Numbers whose prime indices satisfy (length) = (mean), or (sum) = (length)^2.

%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.

%e The terms together with their prime indices begin:

%e     2: {1}

%e     9: {2,2}

%e    10: {1,3}

%e    68: {1,1,7}

%e    78: {1,2,6}

%e    98: {1,4,4}

%e    99: {2,2,5}

%e   105: {2,3,4}

%e   110: {1,3,5}

%e   125: {3,3,3}

%e   328: {1,1,1,13}

%e   444: {1,1,2,12}

%e   558: {1,2,2,11}

%e   620: {1,1,3,11}

%e   783: {2,2,2,10}

%e   812: {1,1,4,10}

%e   870: {1,2,3,10}

%e   966: {1,2,4,9}

%e   988: {1,1,6,8}

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

%t Select[Range[100],Mean[prix[#]]==PrimeOmega[#]&]

%Y Partitions of this type are counted by A364055, without  zeros A206240.

%Y The RHS is A001222.

%Y The LHS is A326567/A326568.

%Y A008284 counts partitions by length, A058398 by mean.

%Y A088529/A088530 gives mean of prime signature A124010.

%Y A112798 lists prime indices, sum A056239.

%Y A124943 counts partitions by low median, high A124944.

%Y A316413 ranks partitions with integer mean, counted by A067538.

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

%Y A363950 ranks partitions with low mean 2, counted by A026905 redoubled.

%Y Cf. A025065, A327473, A327476, A327482, A359889, A363723, A363727, A363729, A363944, A363949.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jul 05 2023