A182863 Members m of A025487 such that, if k appears in m's prime signature, k-1 appears at least as often as k (for any integer k > 1).

1, 2, 6, 12, 30, 60, 210, 360, 420, 1260, 2310, 2520, 4620, 13860, 27720, 30030, 60060, 75600, 138600, 180180, 360360, 510510, 831600, 900900, 1021020, 1801800, 3063060, 6126120, 9699690, 10810800, 15315300, 19399380, 30630600, 37837800

Offset

1,2

Comments

Members m of A025487 such that A181819(m) is also a member of A025487.

If prime signatures are considered as partitions, these are the members of A025487 whose prime signature is conjugate to the prime signature of a member of A181818.

Also the least number with each sorted prime metasignature, where a number's metasignature is the sequence of multiplicities of exponents in its prime factorization. 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}. - Gus Wiseman, May 21 2022

Links

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1444 terms from Amiram Eldar)

Eric Weisstein's World of Mathematics, Conjugate Partition.

Example

The prime signature of 360360 = 2^3*3^2*5*7*11*13 is (3,2,1,1,1,1). 2 appears as many times as 3 in 360360's prime signature, and 1 appears more times than 2. Since 360360 is also a member of A025487, it is a member of this sequence.
From Gus Wiseman, May 21 2022: (Start)
The terms together with their sorted prime signatures and sorted prime metasignatures begin:
      1: {}                -> {}            -> {}
      2: {1}               -> {1}           -> {1}
      6: {1,2}             -> {1,1}         -> {2}
     12: {1,1,2}           -> {1,2}         -> {1,1}
     30: {1,2,3}           -> {1,1,1}       -> {3}
     60: {1,1,2,3}         -> {1,1,2}       -> {1,2}
    210: {1,2,3,4}         -> {1,1,1,1}     -> {4}
    360: {1,1,1,2,2,3}     -> {1,2,3}       -> {1,1,1}
    420: {1,1,2,3,4}       -> {1,1,1,2}     -> {1,3}
   1260: {1,1,2,2,3,4}     -> {1,1,2,2}     -> {2,2}
   2310: {1,2,3,4,5}       -> {1,1,1,1,1}   -> {5}
   2520: {1,1,1,2,2,3,4}   -> {1,1,2,3}     -> {1,1,2}
   4620: {1,1,2,3,4,5}     -> {1,1,1,1,2}   -> {1,4}
  13860: {1,1,2,2,3,4,5}   -> {1,1,1,2,2}   -> {2,3}
  27720: {1,1,1,2,2,3,4,5} -> {1,1,1,2,3}   -> {1,1,3}
  30030: {1,2,3,4,5,6}     -> {1,1,1,1,1,1} -> {6}
  60060: {1,1,2,3,4,5,6}   -> {1,1,1,1,1,2} -> {1,5}
(End)

Mathematica

nn=1000;
r=Table[Sort[Length/@Split[Sort[Last/@If[n==1, {}, FactorInteger[n]]]]], {n, nn}];
Select[Range[nn], !MemberQ[Take[r, #-1], r[[#]]]&] (* Gus Wiseman, May 21 2022 *)

Crossrefs

Intersection of A025487 and A179983.

Subsequence of A129912 and A181826.

Includes all members of A182862.

Positions of first appearances in A353742, unordered version A238747.

A001222 counts prime factors with multiplicity, distinct A001221.

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 distinct prime exponents, counted by A098859.

A181819 gives prime shadow, with an inverse A181821.

A182850 gives frequency depth of prime indices, counted by A225485.

A323014 gives adjusted frequency depth of prime indices, counted by A325280.

Cf. A000040, A055932, A070175, A097318, A304678, A325238, A353507, A353745.

Sequence in context: A331552 A129912 A283477 * A161507 A335711 A032177

Adjacent sequences: A182860 A182861 A182862 * A182864 A182865 A182866

Keyword

nonn

Author

Matthew Vandermast, Jan 14 2011

Status

approved