[seqfan] Re: Is A026477 determined by prime signatures?

[Don Reble]

> A026477... a(1) = 1, a(2) = 2, a(3) = 3; and for n > 3,
> a(n) = smallest number > a(n-1) and not of the form a(i)*a(j)*a(k)
> for 1 <= i < j < k < n.
>
> It seems that if two numbers have the same prime signature (multiset of
> prime exponents) then either both or neither are in the sequence, but I
> can't prove this.

    Just do strong induction on the number of prime factors (sum of
    signature exponents).

> ... prime powers p^r can only be r = {1,2,4,8,15,22...}, ...

    Yes: A026474.
    Also, square-free numbers have 3n+1 prime factors.

    This suggests that A026477 intersect A025487 (least value of each
    signature) would be a worthy sequence. But I don't see how to easily
    recognize those signatures.

       value  signature
           1:
           2:  1
           4:  2
          16:  4
         120:  3  1 1
         210:  1  1 1 1
         216:  3  3
         256:  8
         384:  7  1
        2880:  6  2 1
        6300:  2  2 2 1
        7200:  5  2 2
       15360: 10  1 1
       15552:  6  5
       26880:  8  1 1 1
       27648: 10  3
       32768: 15
       49152: 14  1
       73728: 13  2
       83160:  3  3 1 1 1
      120120:  3  1 1 1 1 1
      189000:  3  3 3 1
      510510:  1  1 1 1 1 1 1
      921600: 12  2 2
     1399680:  7  7 1
     1966080: 17  1 1
     2365440: 11  1 1 1 1
     2822400:  8  2 2 2
     2985984: 12  6
     3440640: 15  1 1 1
     4194304: 22
     4860000:  5  5 4
     5670000:  4  4 4 1
     6291456: 21  1
     6912000: 11  3 3
     9437184: 20  2
    10644480: 10  3 1 1 1
    15375360: 10  1 1 1 1 1
    60466176: 10 10
    65345280:  8  1 1 1 1 1 1
    71663616: 15  7
   117964800: 19  2 2
   127401984: 19  5
   161243136: 13  9
   251658240: 24  1 1
   251942400:  9  9 2
   302776320: 18  1 1 1 1
   361267200: 15  2 2 2
   440401920: 22  1 1 1
   536870912: 29
   805306368: 28  1
   892371480:  3  1 1 1 1 1 1 1 1
  1109908800:  6  2 2 2 2 1
  1207959552: 27  2
  1327104000: 17  4 3
  1968046080: 17  1 1 1 1 1
  4232632320: 11 10 1 1
  6469693230:  1  1 1 1 1 1 1 1 1 1
  9172942848: 22  7
  9932482560: 11  1 1 1 1 1 1 1
10883911680: 12 12 1
12570798240:  5  5 1 1 1 1 1 1
13759414272: 21  8
13783770000:  4  4 4 1 1 1 1
15330615300:  2  2 2 2 2 2 1
16307453952: 26  5
23279477760: 10 10 1 1 1
24461180928: 25  6
32212254720: 31  1 1
32248627200: 16  9 2
38755368960: 25  1 1 1 1
39729690000:  4  4 4 3 1 1
56371445760: 29  1 1 1
68719476736: 36
103079215104: 35  1
114223549440: 10  1 1 1 1 1 1 1 1
154618822656: 34  2
156728328192: 15 14
169869312000: 24  4 3
251909898240: 24  1 1 1 1 1
408410100000:  5  5 5 5
717001084800:  7  2 2 2 2 1 1 1
812665405440: 17 11 1 1
828120733440:  8  1 1 1 1 1 1 1 1 1

-- 
Don Reble  djr at nk.ca