A181818 Products of superprimorials (A006939).

1, 2, 4, 8, 12, 16, 24, 32, 48, 64, 96, 128, 144, 192, 256, 288, 360, 384, 512, 576, 720, 768, 1024, 1152, 1440, 1536, 1728, 2048, 2304, 2880, 3072, 3456, 4096, 4320, 4608, 5760, 6144, 6912, 8192, 8640, 9216, 11520, 12288, 13824, 16384, 17280, 18432, 20736, 23040, 24576, 27648, 32768

Offset

1,2

Comments

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

Subsequence of A025487. A025487(n) belongs to this sequence iff A181815(n) is 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 A182863. - Matthew Vandermast, May 20 2012

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

2, 12, and 360 are all superprimorials (i.e., members of A006939). Therefore, 2*2*12*360 = 17280 is included in the sequence.
From Gus Wiseman, Aug 12 2020 (Start):
The sequence of factorizations (which are unique) begins:
    1 = empty product
    2 = 2
    4 = 2*2
    8 = 2*2*2
   12 = 12
   16 = 2*2*2*2
   24 = 2*12
   32 = 2*2*2*2*2
   48 = 2*2*12
   64 = 2*2*2*2*2*2
   96 = 2*2*2*12
  128 = 2*2*2*2*2*2*2
  144 = 12*12
  192 = 2*2*2*2*12
  256 = 2*2*2*2*2*2*2*2
(End)

Mathematica

Select[Range[100], PrimePi[First/@If[#==1, {}, FactorInteger[#]]]==Range[ PrimeNu[#]]&&LessEqual@@Differences[ Append[Last/@FactorInteger[#], 0]]&] (* Gus Wiseman, Aug 12 2020 *)

Prog

(PARI)
firstdiffs0forward(vec) = { my(v=vector(#vec)); for(n=1, #v, v[n] = vec[n]-if(#v==n, 0, vec[1+n])); (v); };
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)));
isA181818(n) = A353518(n); \\ Antti Karttunen, Apr 28 2022

Crossrefs

A181817 rearranged in numerical order. Also includes all members of A000079, A001021, A006939, A009968, A009992, A066120, A166475, A167448, A181813, A181814, A181816, A182763.

Subsequence of A025487, A055932, A087980, A130091, A181824.

A001013 is the version for factorials.

A336426 is the complement.

A336496 is the version for superfactorials.

A001055 counts factorizations.

A006939 lists superprimorials or Chernoff numbers.

A317829 counts factorizations of superprimorials.

Cf. A022915, A076954, A304686, A325368, A336419, A336420, A336421, A353518 (characteristic function).

Sequence in context: A069519 A363948 A087980 * A363063 A336496 A317804

Adjacent sequences: A181815 A181816 A181817 * A181819 A181820 A181821

Keyword

nonn

Author

Matthew Vandermast, Nov 30 2010

Status

approved