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%I #12 Apr 14 2023 02:13:12
%S 10,30,39,90,98,99,100,115,259,270,273,300,490,495,517,663,665,793,
%T 810,900,1000,1083,1241,1421,1495,1521,1691,1911,2058,2079,2125,2145,
%U 2369,2430,2450,2475,2662,2700,2755,2821,3000,3277,4247,4495,4921,5587,5863,6069
%N Numbers whose prime indices satisfy: (maximum) - (minimum) = (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.
%F A359360(a(n)) = A326844(a(n)).
%F A243055(a(n)) = A061395(a(n)) - A055396(a(n))
%F = A326567(a(n))/A326568(a(n))
%F = A056239(a(n))/A001222(a(n)).
%e The terms together with their prime indices begin:
%e 10: {1,3}
%e 30: {1,2,3}
%e 39: {2,6}
%e 90: {1,2,2,3}
%e 98: {1,4,4}
%e 99: {2,2,5}
%e 100: {1,1,3,3}
%e 115: {3,9}
%e 259: {4,12}
%e 270: {1,2,2,2,3}
%e 273: {2,4,6}
%e 300: {1,1,2,3,3}
%e The prime indices of 490 are {1,3,4,4}, with minimum 1, maximum 4, and mean 3, and 4-1 = 3, so 490 is in the sequence.
%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[100],Max@@prix[#]-Min@@prix[#]==Mean[prix[#]]&]
%o (Python)
%o from itertools import count, islice
%o from sympy import primepi, factorint
%o def A362047_gen(startvalue=2): # generator of terms >= startvalue
%o return filter(lambda n:(primepi(max(f:=factorint(n)))-primepi(min(f)))*sum(f.values())==sum(primepi(i)*j for i, j in f.items()),count(max(startvalue,2)))
%o A362047_list = list(islice(A362047_gen(),20)) # _Chai Wah Wu_, Apr 13 2023
%Y Partitions of this type are counted by A361862.
%Y For minimum instead of mean we have A361908, counted by A118096.
%Y A055396 gives minimum prime index, A061395 maximum.
%Y A112798 list prime indices, length A001222, sum A056239.
%Y A243055 subtracts the least prime index from the greatest.
%Y A326844 gives the diagram complement size of Heinz partition.
%Y Cf. A111907, A237832, A268192, A316413, A326836, A326837, A326846, A359358, A359360, A361855.
%K nonn
%O 1,1
%A _Gus Wiseman_, Apr 11 2023
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