# Prime Factors and A100737

Paul D. Hanna pauldhanna at juno.com
Fri Dec 24 02:45:18 CET 2004

```Paul Barry has some interesting sequences involving the
"inverse modulo 2 binomial transform" of b^n:  A100735-A100741.
These sequences seem to have unusual prime factors.
In particular, consider the prime factors of A100737 (copied below).

When we examine the prime factors of the terms of sequence A100737,
we find that  a(2^n)  appears to be a product of UNIQUE PRIMES ...
at least, this is true for to n=1..8; does this pattern continue for all
n?
I doubt it, yet the behavior of this sequence is interesting to consider.

The prime factors are no doubt related to Fermat primes.

Perhaps someone would like to comment on this number theory inquiry.
Thanks,
Paul Hanna
---------------------------------------------------------------------
Prime factors of first 257 terms of A100737:

3,5,17,257,641,65537,274177,6700417,67280421310721,1238926361552897,
59649589127497217,5704689200685129054721,
93461639715357977769163558199606896584051237541638188580280321

---------------------------------------------------------------------
URL: http://www.research.att.com/projects/OEIS?Anum=A100737
Sequence:
1,3,15,45,255,765,3825,11475,65535,196605,983025,2949075,
16711425,50134275,250671375,752014125,4294967295,
12884901885,64424509425,193273528275,1095216660225,
3285649980675,16428249903375,49284749710125
Name:      Inverse modulo 2 binomial transform of 4^n.
Formula:   a(n)=sum{k=0..n, (-1)^A010060(n-k)*mod(binomial(n,k),2)4^k}.
------------------------------------------------------------
FORMULA.
The formula for A010060(n) is the number of 1's in binary expansion of n
modulo 2.
So, the PARI CODE for A100737 is:

A100737(n)=sum(k=0,n,(-1)^subst(Pol(binary(n-k)),x,1)*(binomial(n,k)%2)*4
^k)

EXAMPLE:
Prime factors of terms shown (n=1..23) are:
3^1
3^1 * 5^1
3^2 * 5^1
3^1 * 5^1 * 17^1
3^2 * 5^1 * 17^1
3^2 * 5^2 * 17^1
3^3 * 5^2 * 17^1
3^1 * 5^1 * 17^1 * 257^1
3^2 * 5^1 * 17^1 * 257^1
3^2 * 5^2 * 17^1 * 257^1
3^3 * 5^2 * 17^1 * 257^1
3^2 * 5^2 * 17^2 * 257^1
3^3 * 5^2 * 17^2 * 257^1
3^3 * 5^3 * 17^2 * 257^1
3^4 * 5^3 * 17^2 * 257^1
3^1 * 5^1 * 17^1 * 257^1 * 65537^1
3^2 * 5^1 * 17^1 * 257^1 * 65537^1
3^2 * 5^2 * 17^1 * 257^1 * 65537^1
3^3 * 5^2 * 17^1 * 257^1 * 65537^1
3^2 * 5^2 * 17^2 * 257^1 * 65537^1
3^3 * 5^2 * 17^2 * 257^1 * 65537^1
3^3 * 5^3 * 17^2 * 257^1 * 65537^1
3^4 * 5^3 * 17^2 * 257^1 * 65537^1

```