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

%I A056153 #17 Jul 25 2024 03:20:10
%S A056153 1,6,30,36,180,210,216,900,1080,1260,1296,2310,5400,6300,6480,7560,
%T A056153 7776,13860,27000,30030,32400,37800,38880,44100,45360,46656,69300,
%U A056153 83160,162000,180180,189000,194400,226800,233280,264600,272160,279936
%N A056153 Leading least prime signatures: a(n) is in A025487 but a(n)/2 is not.
%C A056153 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).
%C A056153 All numbers of the form 2^k1*3^k2*...*p_n^k_n, where k1 = k2 >= ... >= k_n, sorted. - _Robert Israel_, Feb 20 2019
%H A056153 Robert Israel, <a href="/A056153/b056153.txt">Table of n, a(n) for n = 1..10000</a> (first 108 terms from Michel Marcus)
%F A056153 Sum_{n>=1} 1/a(n) = A161360 / 2 = 1.247903257029... . - _Amiram Eldar_, Jul 25 2024
%e A056153 36 is a term because 36 is a member of A025487 but 36/2 = 18 is not.
%e A056153 2520 is a member of A025487 as is 2520/2 = 1260, so 2520 is not a term.
%p A056153 N:= 10^8: # to get all terms <= N
%p A056153 S:= [seq([i,i,6^i],i=0..floor(log[6](N)))]:
%p A056153 Res:= {seq(s[-1],s=S)}:
%p A056153 r:= 6:
%p A056153 for n from 3 do
%p A056153   p:= ithprime(n);
%p A056153   r:= r*p;
%p A056153   if r > N then break fi;
%p A056153   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);
%p A056153   Res:= Res union {seq(s[-1],s=S)};
%p A056153 od:
%p A056153 sort(convert(Res, list)); # _Robert Israel_, Feb 20 2019
%t A056153 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 *)
%o A056153 (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
%o A056153 isok(n) = isli(n) && !isli(n/2); \\ _Michel Marcus_, Feb 20 2019
%Y A056153 Cf. A025487, A062515, A161360.
%K A056153 nonn
%O A056153 1,2
%A A056153 _Alford Arnold_, Jul 30 2000