cp's OEIS Frontend

A382913 Numbers k such that row k of A305936 (a multiset whose multiplicities are the prime indices of k) has a permutation with all distinct run-lengths.

%I A382913 #15 May 08 2025 19:42:04
%S A382913 1,2,3,5,6,7,10,11,13,14,15,17,19,21,22,23,25,26,29,30,31,33,34,35,37,
%T A382913 38,39,41,42,43,46,47,49,51,53,55,57,58,59,61,62,65,66,67,69,70,71,73,
%U A382913 74,77,78,79,82,83,85,86,87,89,91,93,94,95,97,101,102,103
%N A382913 Numbers k such that row k of A305936 (a multiset whose multiplicities are the prime indices of k) has a permutation with all distinct run-lengths.
%C A382913 This described multiset (row n of A305936, Heinz number A181821) is generally not the same as the multiset of prime indices of n (A112798). For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.
%C A382913 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 A382913 The terms, prime indices, and corresponding multisets begin:
%e A382913    1:    {} {}
%e A382913    2:   {1} {1}
%e A382913    3:   {2} {1,1}
%e A382913    5:   {3} {1,1,1}
%e A382913    6: {1,2} {1,1,2}
%e A382913    7:   {4} {1,1,1,1}
%e A382913   10: {1,3} {1,1,1,2}
%e A382913   11:   {5} {1,1,1,1,1}
%e A382913   13:   {6} {1,1,1,1,1,1}
%e A382913   14: {1,4} {1,1,1,1,2}
%e A382913   15: {2,3} {1,1,1,2,2}
%e A382913   17:   {7} {1,1,1,1,1,1,1}
%e A382913   19:   {8} {1,1,1,1,1,1,1,1}
%e A382913   21: {2,4} {1,1,1,1,2,2}
%e A382913   22: {1,5} {1,1,1,1,1,2}
%e A382913   23:   {9} {1,1,1,1,1,1,1,1,1}
%e A382913   25: {3,3} {1,1,1,2,2,2}
%e A382913   26: {1,6} {1,1,1,1,1,1,2}
%t A382913 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&, If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_} :> Table[PrimePi[p],{k}]]]]];
%t A382913 lasQ[y_]:=Select[Permutations[y], UnsameQ@@Length/@Split[#]&]!={};
%t A382913 Select[Range[100],lasQ@*nrmptn]
%Y A382913 Look-and-Say partitions are counted by A239455, ranks A351294.
%Y A382913 Non-Look-and-Say partitions are counted by A351293, ranks A351295.
%Y A382913 For prime indices instead of signature we have A351294, conjugate A381432.
%Y A382913 The Look-and-Say partition of n is listed by A381440, rank A048767.
%Y A382913 The complement is A382912.
%Y A382913 For equal run-lengths we have the complement of A382914, see A382858, A382879, A382915.
%Y A382913 A044813 lists numbers whose binary expansion has distinct run-lengths.
%Y A382913 A055396 gives least prime index, greatest A061395.
%Y A382913 A056239 adds up prime indices, row sums of A112798.
%Y A382913 A329739 counts compositions with distinct run-lengths, ranks A351596.
%Y A382913 A381431 ranks section-sum partition, listed by A381436.
%Y A382913 Cf. A000720, A001221, A001222, A140690, A181821, A305936, A335125, A382525, A382771, A382773, A382775.
%Y A382913 Cf. A381636, A381717, A381871, A382876.
%K A382913 nonn
%O A382913 1,2
%A A382913 _Gus Wiseman_, Apr 12 2025