cp's OEIS Frontend

A360247 Numbers for which the prime indices have the same mean as the distinct prime indices.

%I A360247 #7 May 22 2023 05:43:18
%S A360247 1,2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,19,21,22,23,25,26,27,29,30,31,
%T A360247 32,33,34,35,36,37,38,39,41,42,43,46,47,49,51,53,55,57,58,59,61,62,64,
%U A360247 65,66,67,69,70,71,73,74,77,78,79,81,82,83,85,86,87,89,90,91,93,94,95,97,100,101,102,103,105,106,107,109,110,111,113,114,115,118,119,121,122,123,125,127,128,129,130
%N A360247 Numbers for which the prime indices have the same mean as the distinct prime indices.
%C A360247 First differs from A072774 in having 90.
%C A360247 First differs from A242414 in lacking 126.
%C A360247 Includes all squarefree numbers and perfect powers.
%C A360247 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 A360247 The prime indices of 900 are {3,3,2,2,1,1} with mean 2, and the distinct prime indices are {1,2,3} also with mean 2, so 900 is in the sequence.
%p A360247 isA360247 := proc(n)
%p A360247     local ifs,pidx,pe,meanAll,meanDist ;
%p A360247     if n = 1 then
%p A360247         return true ;
%p A360247     end if ;
%p A360247     ifs := ifactors(n)[2] ;
%p A360247     # list of prime indices with multiplicity
%p A360247     pidx := [] ;
%p A360247     for pe in ifs do
%p A360247         [numtheory[pi](op(1,pe)),op(2,pe)] ;
%p A360247         pidx := [op(pidx),%] ;
%p A360247     end do:
%p A360247     meanAll := add(op(1,pe)*op(2,pe),pe=pidx) / add(op(2,pe),pe=pidx) ;
%p A360247     meanDist := add(op(1,pe),pe=pidx) / nops(pidx) ;
%p A360247     if meanAll = meanDist then
%p A360247         true;
%p A360247     else
%p A360247         false;
%p A360247     end if;
%p A360247 end proc:
%p A360247 for n from 1 to 130 do
%p A360247     if isA360247(n) then
%p A360247         printf("%d,",n) ;
%p A360247     end if;
%p A360247 end do: # _R. J. Mathar_, May 22 2023
%t A360247 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A360247 Select[Range[100],Mean[prix[#]]==Mean[Union[prix[#]]]&]
%Y A360247 Signature instead of parts: A324570, counted by A114638.
%Y A360247 Signature instead of distinct parts: A359903, counted by A360068.
%Y A360247 These partitions are counted by A360243.
%Y A360247 The complement is A360246, counted by A360242.
%Y A360247 For median instead of mean the complement is A360248, counted by A360244.
%Y A360247 For median instead of mean we have A360249, counted by A360245.
%Y A360247 For greater instead of equal mean we have A360252, counted by A360250.
%Y A360247 For lesser instead of equal mean we have A360253, counted by A360251.
%Y A360247 A008284 counts partitions by number of parts, distinct A116608.
%Y A360247 A058398 counts partitions by mean, also A327482.
%Y A360247 A088529/A088530 gives mean of prime signature (A124010).
%Y A360247 A112798 lists prime indices, length A001222, sum A056239.
%Y A360247 A316413 = numbers whose prime indices have integer mean, distinct A326621.
%Y A360247 A326567/A326568 gives mean of prime indices.
%Y A360247 A326619/A326620 gives mean of distinct prime indices.
%Y A360247 Cf. A000975, A051293, A067340, A067538, A360005, A360241.
%K A360247 nonn
%O A360247 1,2
%A A360247 _Gus Wiseman_, Feb 07 2023