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A181818 Products of superprimorials (A006939).

%I A181818 #38 Jul 17 2023 07:59:52
%S A181818 1,2,4,8,12,16,24,32,48,64,96,128,144,192,256,288,360,384,512,576,720,
%T A181818 768,1024,1152,1440,1536,1728,2048,2304,2880,3072,3456,4096,4320,4608,
%U A181818 5760,6144,6912,8192,8640,9216,11520,12288,13824,16384,17280,18432,20736,23040,24576,27648,32768
%N A181818 Products of superprimorials (A006939).
%C A181818 Sorted list of positive integers with a factorization Product p(i)^e(i) such that (e(1) - e(2)) >= (e(2) - e(3)) >= ... >= (e(k-1) - e(k)) >= e(k), with k = A001221(n), and p(k) = A006530(n) = A000040(k), i.e., the prime factors p(1) .. p(k) must be consecutive primes from 2 onward. - Comment clarified by _Antti Karttunen_, Apr 28 2022
%C A181818 Subsequence of A025487. A025487(n) belongs to this sequence iff A181815(n) is a member of A025487.
%C A181818 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 A182863. - _Matthew Vandermast_, May 20 2012
%H A181818 Amiram Eldar, <a href="/A181818/b181818.txt">Table of n, a(n) for n = 1..10000</a>
%e A181818 2, 12, and 360 are all superprimorials (i.e., members of A006939). Therefore, 2*2*12*360 = 17280 is included in the sequence.
%e A181818 From _Gus Wiseman_, Aug 12 2020 (Start):
%e A181818 The sequence of factorizations (which are unique) begins:
%e A181818     1 = empty product
%e A181818     2 = 2
%e A181818     4 = 2*2
%e A181818     8 = 2*2*2
%e A181818    12 = 12
%e A181818    16 = 2*2*2*2
%e A181818    24 = 2*12
%e A181818    32 = 2*2*2*2*2
%e A181818    48 = 2*2*12
%e A181818    64 = 2*2*2*2*2*2
%e A181818    96 = 2*2*2*12
%e A181818   128 = 2*2*2*2*2*2*2
%e A181818   144 = 12*12
%e A181818   192 = 2*2*2*2*12
%e A181818   256 = 2*2*2*2*2*2*2*2
%e A181818 (End)
%t A181818 Select[Range[100],PrimePi[First/@If[#==1,{}, FactorInteger[#]]]==Range[ PrimeNu[#]]&&LessEqual@@Differences[ Append[Last/@FactorInteger[#],0]]&] (* _Gus Wiseman_, Aug 12 2020 *)
%o A181818 (PARI)
%o A181818 firstdiffs0forward(vec) = { my(v=vector(#vec)); for(n=1,#v,v[n] = vec[n]-if(#v==n,0,vec[1+n])); (v); };
%o A181818 A353518(n) = if(1==n,1,my(f=factor(n), len=#f~); if(primepi(f[len,1])!=len, return(0), my(diffs=firstdiffs0forward(f[,2])); for(i=1,#diffs-1,if(diffs[i+1]>diffs[i],return(0))); (1)));
%o A181818 isA181818(n) = A353518(n); \\ _Antti Karttunen_, Apr 28 2022
%Y A181818 A181817 rearranged in numerical order. Also includes all members of A000079, A001021, A006939, A009968, A009992, A066120, A166475, A167448, A181813, A181814, A181816, A182763.
%Y A181818 Subsequence of A025487, A055932, A087980, A130091, A181824.
%Y A181818 A001013 is the version for factorials.
%Y A181818 A336426 is the complement.
%Y A181818 A336496 is the version for superfactorials.
%Y A181818 A001055 counts factorizations.
%Y A181818 A006939 lists superprimorials or Chernoff numbers.
%Y A181818 A317829 counts factorizations of superprimorials.
%Y A181818 Cf. A022915, A076954, A304686, A325368, A336419, A336420, A336421, A353518 (characteristic function).
%K A181818 nonn
%O A181818 1,2
%A A181818 _Matthew Vandermast_, Nov 30 2010