cp's OEIS Frontend

A382878 Set of positions of first appearances in A382857 (permutations of prime indices with equal run-lengths).

%I A382878 #6 Apr 10 2025 23:17:13
%S A382878 1,6,24,30,36,180,210,360,420,720,1080,1260,1800,2160,2310,2520,3600,
%T A382878 4620,5040,5400,6300,7560,10800,12600,13860,15120,21600,25200,25920,
%U A382878 27000,27720,30030,32400,37800,44100,45360,46656,50400,54000,55440,60060,60480,64800
%N A382878 Set of positions of first appearances in A382857 (permutations of prime indices with equal run-lengths).
%C A382878 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, sum A056239.
%e A382878 The permutations for n = 6, 720, 36, 25920, 30:
%e A382878   (1,2)  (1,2,1,2,1,3,1)  (1,1,2,2)  (1,2,1,2,1,2,1,2,1,3,1)  (1,2,3)
%e A382878   (2,1)  (1,2,1,3,1,2,1)  (1,2,1,2)  (1,2,1,2,1,2,1,3,1,2,1)  (1,3,2)
%e A382878          (1,3,1,2,1,2,1)  (2,1,2,1)  (1,2,1,2,1,3,1,2,1,2,1)  (2,1,3)
%e A382878                           (2,2,1,1)  (1,2,1,3,1,2,1,2,1,2,1)  (2,3,1)
%e A382878                                      (1,3,1,2,1,2,1,2,1,2,1)  (3,1,2)
%e A382878                                                               (3,2,1)
%e A382878 The terms together with their prime indices begin:
%e A382878       1: {}
%e A382878       6: {1,2}
%e A382878      24: {1,1,1,2}
%e A382878      30: {1,2,3}
%e A382878      36: {1,1,2,2}
%e A382878     180: {1,1,2,2,3}
%e A382878     210: {1,2,3,4}
%e A382878     360: {1,1,1,2,2,3}
%e A382878     420: {1,1,2,3,4}
%e A382878     720: {1,1,1,1,2,2,3}
%e A382878    1080: {1,1,1,2,2,2,3}
%e A382878    1260: {1,1,2,2,3,4}
%e A382878    1800: {1,1,1,2,2,3,3}
%e A382878    2160: {1,1,1,1,2,2,2,3}
%e A382878    2310: {1,2,3,4,5}
%e A382878    2520: {1,1,1,2,2,3,4}
%e A382878    3600: {1,1,1,1,2,2,3,3}
%t A382878 y=Table[Length[Select[Permutations[Join@@ConstantArray@@@FactorInteger[n]],SameQ@@Length/@Split[#]&]],{n,0,1000}];
%t A382878 fip[y_]:=Select[Range[Length[y]],!MemberQ[Take[y,#-1],y[[#]]]&];
%t A382878 fip[Rest[y]]
%Y A382878 Positions of first appearances in A382857 (zeros A382879), by signature A382858.
%Y A382878 For distinct run-lengths we have A382772, firsts of A382771 (by signature A382773).
%Y A382878 A140690 lists numbers whose binary expansion has equal run-lengths, distinct A044813.
%Y A382878 A056239 adds up prime indices, row sums of A112798.
%Y A382878 A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
%Y A382878 A329738 counts compositions with equal run-lengths, ranks A353744.
%Y A382878 A329739 counts compositions with distinct run-lengths, ranks A351596.
%Y A382878 A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
%Y A382878 Cf. A000720, A001221, A001222, A003242, A048767, A098859, A130091, A238130, A305936, A351013, A351202, A382876.
%K A382878 nonn
%O A382878 1,2
%A A382878 _Gus Wiseman_, Apr 09 2025