A331962 Triangle read by rows: row j lists j primes by ascending n bit bientropy (n >= 8).

2, 3, 2, 5, 3, 2, 5, 3, 2, 7, 5, 3, 2, 7, 11, 5, 3, 2, 7, 11, 13, 17, 5, 3, 2, 7, 11, 13, 17, 5, 3, 2, 7, 11, 19, 13, 17, 5, 3, 23, 2, 7, 11, 19, 13, 17, 5, 3, 23, 29, 2, 7, 11, 19, 13, 17, 5, 3, 23, 29, 2, 7, 31, 11, 19, 13, 17, 5, 3, 23, 29, 2, 7, 31, 11, 19, 37, 13

Offset

1,1

Comments

Primes are sorted by the relative order and disorder of their n binary digits as measured by the bientropy function. When two or more primes have the same bientropy, their natural order is preserved.

References

G. J. Croll, Bientropy - the measurement and algebras of order and disorder in finite binary strings. Scientific essays in honor of H. Pierre Noyes on the occasion of his 90th Birthday, World Scientific, 2014, pages 48-64.

Links

Table of n, a(n) for n=1..78.

Grenville J. Croll, BiEntropy, TriEntropy and Primality, arXiv:1912.08051 [cs.OH], 2019.

Grenville J. Croll, BiEntropy, TriEntropy and Primality, Entropy 2020, 22, 311.

Grenville J. Croll, BiEntropy_TriEntropy_and_Primality.zip, Figshare Dataset, 2020.

Formula

Given a binary string s = s1...sn (where in this application n >= 8), there are n-1 binary derivatives of s, Dk(s), 1 <= k <= n-1.

The first binary derivative of s, D1(s), is the binary string of length n-1 formed by XORing adjacent pairs of digits of s.

We refer to the k-th derivative of s, Dk(s), as the binary derivative of Dk-1(s).

p(k) is the proportion of 1's in Dk.

D0 = s;

0*log_2(0) is defined to be 0;

C = 1/(2^(n-1)-1);

D = Sum_{k=0..n-2}(-p(k)*log_2(p(k))-(1-p(k))*log_2(1-p(k)))*2^k.

bientropy(s) = C*D;

Example

For example, the 8-bit bientropy of 17 (a Fermat prime) is 0.0534 as, in binary, it is a periodic binary number - 00010001. The bientropy of 13 is 0.9532 as its binary digits - 00001101 are not periodic. Each row j reveals where the j-th prime sits relative to the rest given its bientropy.
Triangle begins:
2,
3, 2,
5, 3, 2,
5, 3, 2, 7,
5, 3, 2, 7, 11
5, 3, 2, 7, 11, 13
17, 5, 3, 2, 7, 11, 13
17, 5, 3, 2, 7, 11, 19, 13
17, 5, 3, 23, 2, 7, 11, 19, 13
17, 5, 3, 23, 29, 2, 7, 11, 19, 13
17, 5, 3, 23, 29, 2, 7, 31, 11, 19, 13
17, 5, 3, 23, 29, 2, 7, 31, 11, 19, 37, 13
...

Prog

(Excel) See Grenville J. Croll Figshare link.

Crossrefs

Cf. A000040, A019434.

Sequence in context: A306252 A112764 A108728 * A302170 A049805 A104887

Adjacent sequences: A331959 A331960 A331961 * A331963 A331964 A331965

Keyword

tabl,easy,nonn,base

Author

Grenville J. Croll, Feb 02 2020

Status

approved