Seq.s A050150 & A062090. (2)

[Klaus Brockhaus]

Robert G. Wilson v schrieb:
> 
> Sequence Fanatics,
> 
>         A062090 is a(1) = 1, a(n)= smallest odd number which does not
> divide the product of all previous terms.
> 
>         A050150 is Odd numbers with prime number of divisors.
> 
>         They both begin: 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 37,
> 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79,
> 81, 83, 89, 97, 101, 103, 107, 109, ..., .
> 
>         In the comments section of A050150 it says that "This sequence
> differs from A062090 by one term, "81", so far." I do not know who said
> this nor do I care, but it is wrong. The two sequences match except for
> the initial 1 which is needed to begin A062090 up to 729. It appears in
> A050150 but not in A062090. Then 6561 appears in A062090 but not in
> A050150. I'll push this further. Any comments?
> 
> Sincerely,
> 
> Robert G. "Bob" Wilson, v


Let p stand for an odd prime.
Using the characterization, communicated by David W. Wilson,
	p^(q-1) where q is prime except for 2 or a Fermat prime
for the numbers which are in A050150 but not in A062090, I obtained the
sequence (of terms < 10^12)

729, 15625, 59049, 117649, 531441, 1771561, 4826809, 9765625, 24137569,
47045881, 148035889, 244140625, 282475249, 387420489, 594823321,
887503681, 2565726409, 4750104241, 6321363049, 10779215329, 13841287201,
22164361129, 25937424601, 31381059609, 42180533641, 51520374361,
90458382169, 128100283921, 137858491849, 151334226289, 243087455521,
326940373369, 496981290961, 832972004929, ...

Using the characterization, also communicated by David W. Wilson,
	p^2^n where 2^n+1 is neither 2 nor a Fermat prime
for the numbers (except for the initial 1) which are in A062090 but not
in A050150, I obtained the sequence (of terms < 10^17)

6561, 390625, 5764801, 214358881, 815730721, 6975757441, 16983563041,
78310985281, 500246412961, 852891037441, 3512479453921, 7984925229121,
11688200277601, 23811286661761, 62259690411361, 146830437604321,
191707312997281, 406067677556641, 645753531245761, 806460091894081,
1517108809906561, 1853020188851841, 2252292232139041, 3936588805702081,
7837433594376961, 10828567056280801, 12667700813876161,
17181861798319201, 19925626416901921, 26584441929064321,
67675234241018881, 86730203469006241, ...
(coincides in the beginning with p^8 (p > 2), first divergence is at
3^32 = 1853020188851841).

David, thanks for the message!

Klaus Brockhaus