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A382912 Numbers k such that row k of A305936 (a multiset whose multiplicities are the prime indices of k) has no permutation with all distinct run-lengths.

%I A382912 #15 May 08 2025 19:42:09
%S A382912 4,8,9,12,16,18,20,24,27,28,32,36,40,44,45,48,50,52,54,56,60,63,64,68,
%T A382912 72,75,76,80,81,84,88,90,92,96,98,99,100,104,108,112,116,117,120,124,
%U A382912 125,126,128,132,135,136,140,144,148,150,152,153,156,160,162,164
%N A382912 Numbers k such that row k of A305936 (a multiset whose multiplicities are the prime indices of k) has no permutation with all distinct run-lengths.
%C A382912 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 A382912 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 A382912 The terms, prime indices, and corresponding multisets begin:
%e A382912    4:       {1,1} {1,2}
%e A382912    8:     {1,1,1} {1,2,3}
%e A382912    9:       {2,2} {1,1,2,2}
%e A382912   12:     {1,1,2} {1,1,2,3}
%e A382912   16:   {1,1,1,1} {1,2,3,4}
%e A382912   18:     {1,2,2} {1,1,2,2,3}
%e A382912   20:     {1,1,3} {1,1,1,2,3}
%e A382912   24:   {1,1,1,2} {1,1,2,3,4}
%e A382912   27:     {2,2,2} {1,1,2,2,3,3}
%e A382912   28:     {1,1,4} {1,1,1,1,2,3}
%e A382912   32: {1,1,1,1,1} {1,2,3,4,5}
%e A382912   36:   {1,1,2,2} {1,1,2,2,3,4}
%e A382912   40:   {1,1,1,3} {1,1,1,2,3,4}
%e A382912   44:     {1,1,5} {1,1,1,1,1,2,3}
%e A382912   45:     {2,2,3} {1,1,1,2,2,3,3}
%e A382912   48: {1,1,1,1,2} {1,1,2,3,4,5}
%e A382912   50:     {1,3,3} {1,1,1,2,2,2,3}
%e A382912   52:     {1,1,6} {1,1,1,1,1,1,2,3}
%t A382912 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{}, Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_} :> Table[PrimePi[p],{k}]]]]];
%t A382912 lasQ[y_]:=Select[Permutations[y], UnsameQ@@Length/@Split[#]&]!={};
%t A382912 Select[Range[100],Not@*lasQ@*nrmptn]
%Y A382912 The Look-and-Say partition is ranked by A048767, listed by A381440.
%Y A382912 Look-and-Say partitions are counted by A239455, ranks A351294.
%Y A382912 Non-Look-and-Say partitions are counted by A351293.
%Y A382912 For prime indices instead of signature we have A351295, conjugate A381433.
%Y A382912 The complement is A382913.
%Y A382912 For equal instead of distinct run-lengths we have A382914, see A382858, A382879, A382915.
%Y A382912 A056239 adds up prime indices, row sums of A112798.
%Y A382912 A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
%Y A382912 A381431 lists the section-sum partition of n, ranks A381436, union A381432.
%Y A382912 Cf. A000720, A001221, A001222, A181821, A305936, A335125, A351202, A382525, A382771, A382773, A382775, A382857.
%Y A382912 Cf. A381636, A381717, A381871, A382876.
%K A382912 nonn
%O A382912 1,1
%A A382912 _Gus Wiseman_, Apr 12 2025