cp's OEIS Frontend

A363951 Numbers whose prime indices satisfy (length) = (mean), or (sum) = (length)^2.

%I A363951 #7 Jul 06 2023 08:55:16
%S A363951 2,9,10,68,78,98,99,105,110,125,328,444,558,620,783,812,870,966,988,
%T A363951 1012,1035,1150,1156,1168,1197,1254,1326,1330,1425,1521,1666,1683,
%U A363951 1690,1704,1785,1870,1911,2002,2125,2145,2275,2401,2412,2541,2662,2680,2695,3025
%N A363951 Numbers whose prime indices satisfy (length) = (mean), or (sum) = (length)^2.
%C A363951 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 A363951 The terms together with their prime indices begin:
%e A363951     2: {1}
%e A363951     9: {2,2}
%e A363951    10: {1,3}
%e A363951    68: {1,1,7}
%e A363951    78: {1,2,6}
%e A363951    98: {1,4,4}
%e A363951    99: {2,2,5}
%e A363951   105: {2,3,4}
%e A363951   110: {1,3,5}
%e A363951   125: {3,3,3}
%e A363951   328: {1,1,1,13}
%e A363951   444: {1,1,2,12}
%e A363951   558: {1,2,2,11}
%e A363951   620: {1,1,3,11}
%e A363951   783: {2,2,2,10}
%e A363951   812: {1,1,4,10}
%e A363951   870: {1,2,3,10}
%e A363951   966: {1,2,4,9}
%e A363951   988: {1,1,6,8}
%t A363951 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A363951 Select[Range[100],Mean[prix[#]]==PrimeOmega[#]&]
%Y A363951 Partitions of this type are counted by A364055, without  zeros A206240.
%Y A363951 The RHS is A001222.
%Y A363951 The LHS is A326567/A326568.
%Y A363951 A008284 counts partitions by length, A058398 by mean.
%Y A363951 A088529/A088530 gives mean of prime signature A124010.
%Y A363951 A112798 lists prime indices, sum A056239.
%Y A363951 A124943 counts partitions by low median, high A124944.
%Y A363951 A316413 ranks partitions with integer mean, counted by A067538.
%Y A363951 A326622 counts factorizations with integer mean, strict A328966.
%Y A363951 A363950 ranks partitions with low mean 2, counted by A026905 redoubled.
%Y A363951 Cf. A025065, A327473, A327476, A327482, A359889, A363723, A363727, A363729, A363944, A363949.
%K A363951 nonn
%O A363951 1,1
%A A363951 _Gus Wiseman_, Jul 05 2023