A353742 Sorted prime metasignature of n.

1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 3, 1, 1

Offset

1,5

Comments

The prime metasignature counts the multiplicities of each value in the prime signature of n. For example, 2520 has prime indices {1,1,1,2,2,3,4}, sorted prime signature {1,1,2,3}, and sorted prime metasignature {1,1,2}.

Links

Table of n, a(n) for n=1..87.

Example

The prime indices, sorted prime signatures, and sorted prime metasignatures of selected n:
      n = 1: {}             -> {}         -> {}
      n = 2: {1}            -> {1}        -> {1}
      n = 6: {1,2}          -> {1,1}      -> {2}
     n = 12: {1,1,2}        -> {1,2}      -> {1,1}
     n = 30: {1,2,3}        -> {1,1,1}    -> {3}
     n = 60: {1,1,2,3}      -> {1,1,2}    -> {1,2}
    n = 210: {1,2,3,4}      -> {1,1,1,1}  -> {4}
    n = 360: {1,1,1,2,2,3}  -> {1,2,3}    -> {1,1,1}

Mathematica

Join@@Table[Sort[Length/@Split[Sort[Last/@If[n==1, {}, FactorInteger[n]]]]], {n, 100}]

Crossrefs

Row-sums are A001221.

Row-lengths are A071625.

Positions of first appearances are A182863.

This is the sorted version of A238747.

Row-products are A353507.

A001222 counts prime factors with multiplicity.

A003963 gives product of prime indices.

A005361 gives product of prime signature, firsts A353500 (sorted A085629).

A056239 adds up prime indices, row sums of A112798 and A296150.

A124010 gives prime signature, sorted A118914.

A130091 lists numbers with strict signature, counted by A098859.

A181819 gives prime shadow, with an inverse A181821.

Cf. A000040, A000720, A070175, A097318, A182850, A304678, A325238, A353503.

Sequence in context: A354057 A143898 A332636 * A238747 A101873 A336691

Adjacent sequences: A353739 A353740 A353741 * A353743 A353744 A353745

Keyword

nonn,tabf

Author

Gus Wiseman, May 20 2022

Status

approved