A056153 Leading least prime signatures: a(n) is in A025487 but a(n)/2 is not.

1, 6, 30, 36, 180, 210, 216, 900, 1080, 1260, 1296, 2310, 5400, 6300, 6480, 7560, 7776, 13860, 27000, 30030, 32400, 37800, 38880, 44100, 45360, 46656, 69300, 83160, 162000, 180180, 189000, 194400, 226800, 233280, 264600, 272160, 279936

Offset: 1

Author

Alford Arnold, Jul 30 2000

Keywords

• nonn

Comments

Values of A025487 can be mapped to the numeric partitions. In a similar way, values of a(n) can be mapped to the cyclic partitions by noting that the factorization vector begins (k, k, ...). E.g. 1260 = 2*2*3*3*5*7 yielding the vector (2,2,1,1).

All numbers of the form 2^k1*3^k2*...*p_n^k_n, where k1 = k2 >= ... >= k_n, sorted. - Robert Israel, Feb 20 2019

Examples

			36 is a term because 36 is a member of A025487 but 36/2 = 18 is not.
2520 is a member of A025487 as is 2520/2 = 1260, so 2520 is not a term.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000 (first 108 terms from Michel Marcus)

Crossrefs

Cf. A025487, A062515, A161360.

Programs

Maple

N:= 10^8: # to get all terms <= N
S:= [seq([i,i,6^i],i=0..floor(log[6](N)))]:
Res:= {seq(s[-1],s=S)}:
r:= 6:
for n from 3 do
  p:= ithprime(n);
  r:= r*p;
  if r > N then break fi;
  S:= map(t ->seq([op(t[1..-2]),i,t[-1]*p^i],i=1..min(t[-2], floor(log[p](N/t[-1])))), S);
  Res:= Res union {seq(s[-1],s=S)};
od:
sort(convert(Res, list)); # Robert Israel, Feb 20 2019

Mathematica

max = 300000; ss = {}; A025487 = Join[{1}, Reap[ Do[s = Sort[FactorInteger[n][[All, 2]]]; If[FreeQ[ss, s], AppendTo[ss, s]; Sow[n]], {n, 2, max}]][[2, 1]]]; Select[A025487, FreeQ[A025487, #/2] &] (* Jean-François Alcover, Jul 11 2012 *)

PARI

isli(n) = if(n==1, return(1)); if (frac(n), return (0)); my(f = factor(n)); f[#f~, 1] == prime(#f~) && vecsort(f[, 2], , 4) == f[, 2]; \\ A025487
isok(n) = isli(n) && !isli(n/2); \\ Michel Marcus, Feb 20 2019

Formula

Sum_{n>=1} 1/a(n) = A161360 / 2 = 1.247903257029... . - Amiram Eldar, Jul 25 2024