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

[Charles Greathouse]

I also haven't found a good way of discovering which prime signatures are
in the sequence. In principle this is combanatorial but I don't know of a
good algorithm.

Charles Greathouse
Case Western Reserve University

On Fri, Aug 26, 2016 at 7:17 AM, Don Reble <djr at nk.ca> wrote:

> 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
>
>
>
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