[seqfan] Re: Ramanujan sequence A067128

David Corneth davidacorneth at gmail.com
Fri May 13 17:01:07 CEST 2016


Hi Vladimir et all,

I've also sumbitted a sequence relating to A067128. It's A273014, by the
name
Least k such that A067128(k) is divisible by n.

I'd like to put another sequence:
Least k sucht that A067128(m) is divisible by n for all m>=k, but I find it
a bit difficult to find data for this one.

It's based on the idea of a prime signature. I thought maybe that's a way
to get more insight in the sequence and maybe to see if A067128 = A034287.
A prime signature of a number n is a vector of size primepi(maxp(n)) where
maxp(n) is the highest prime dividing n, and primepi(p) counts the primes
upto p. The elements with index i of the vector are the exponents of the
i-th prime in the factorization of n.
To give an example, The prime signature of 264 is [3, 1, 0, 0, 1]. The
highest prime dividing 264 is 11. Primepi(11) = 5 so the signature has 5
elements.
The exponent of the first prime, 2, is 3 in the factorization of 264, so
the first element of the sig. is 3.
The exponent of the second prime, 3, is 1 in the factorization of 264, so
the second element of the sig. is 1.
The exponent of the third prime, 5, is 0 in the factorization of 264, so
the third element of the sig. is 0.
The exponent of the fourth prime, 7, is 0 in the factorization of 264, so
the fourth element of the sig. is 0.
The exponent of the fourth prime, 11, is 0 in the factorization of 264, so
the fourth element of the sig. is 1.

I put the prime signatures of the first 555 elements of A067128 in Excel
and colored the cells according to the elements in the signatures. This
gives a cool colorscheme!

Best,
David


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