A382914 Numbers k such that it is not possible to permute a multiset whose multiplicities are the prime indices of k so that the run-lengths are all equal.

10, 14, 22, 26, 28, 33, 34, 38, 39, 44, 46, 51, 52, 55, 57, 58, 62, 66, 68, 69, 74, 76, 78, 82, 85, 86, 87, 88, 92, 93, 94, 95, 102, 104, 106, 111, 114, 115, 116, 118, 119, 122, 123, 124, 129, 130, 134, 136, 138, 141, 142, 145, 146, 148, 152, 153, 155, 156

Offset: 1

Author

Gus Wiseman, Apr 09 2025

Keywords

• nonn

Comments

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.

Examples

			The terms together with their prime indices begin:
  10: {1,3}
  14: {1,4}
  22: {1,5}
  26: {1,6}
  28: {1,1,4}
  33: {2,5}
  34: {1,7}
  38: {1,8}
  39: {2,6}
  44: {1,1,5}
  46: {1,9}
  51: {2,7}
  52: {1,1,6}
  55: {3,5}
  57: {2,8}
  58: {1,10}
  62: {1,11}
  66: {1,2,5}
		

Crossrefs

For anti-run permutations we have A335126, complement A335127.

Zeros of A382858, anti-run A335125.

For prime indices instead of signature we have A382879, counted by A382915.

For distinct run-lengths we have A382912 (zeros of A382773), complement A382913.

A003963 gives product of prime indices.

A056239 adds up prime indices, row sums of A112798.

A140690 lists numbers whose binary expansion has equal run-lengths, distinct A044813.

A304442 counts partitions with equal run-sums, ranks A353833.

A164707 lists numbers whose binary form has equal runs of ones, distinct A328592.

A329738 counts compositions with equal run-lengths, ranks A353744.

A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.

Cf. A382857 (firsts A382878), A382771 (firsts A382772).

Cf. A000720, A000961, A001221, A001222, A048767, A181821, A305936, A351013, A382877.

Programs

Mathematica

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],Select[Permutations[nrmptn[#]],SameQ@@Length/@Split[#]&]=={}&]