Proposed seq: Smallest prime with exactly 2n+1 ones when written in base 3

[Richard Mathar]

Richard Mathar is correct in every respect.

Combining his extensions (and his bounds are good formulae):

>Adding a few more terms we get
3,13,283,1093,22963,259159,797161,19929037,150663523
>
>  And because the smallest number with n ones in base 3 is
>  3^0+3^1+3^2+...+3^(n-1)=(3^n-1)/2=A003462(n), we have a lower bound
>
>  a(n) >= A003462(2n+1) = A096053(n).
>
>  Definition could be rephrased as:
>
>  a(n) := {min A000040(i): A073780(i)=2n+1}.
>  ...
>  We also have "smallest prime with exactly n ones in base 4"
>  7,5,89,277,1109,5477,17749,70997,349529,
>
>  "smallest prime with exactly n ones in base 5":
>  5,41,31,1031,3881,19541,19531
>
>  "smallest prime with exactly n ones in base 6":
>  11,7,43,1549,9337,56131,55987

All this could be captured in a single sequence, using an array (by
antidiagonals):

A[k,n] = smallest prime with exactly n ones in base k, or 0 if no such prime.

A[1,n] = n * the characteristic function of primes (i.e. in unary, an
integer has n 1's and is prime iff n = that prime)

A[10,n] = A037055  Smallest prime containing exactly n 1's.

The array begins:
========================================
........|.n=0.|.n=1.|.n=2.|..n=3.|....n=4.|......n=5.|.........n=6.|.........n=7.|.......n=8.|n=9.|
k=1..|...0...|....0...|...2...|.....3...|.......0...|.........5...|...........0...|............7...|................0..|...0
k=2..|...0...|....2...|...3...|.....7...|.....23...|.......31...|........311..|........127...|.............383.|....991..|...2039
k=3..|...2...|....3...|...0...|...13...|.......0...|.....283...|............0..|.......1093...|..............0...|....22963..
k=4..|...0...|....7...|...5...|...89...|...277...|...1109..|......5477..|.......17749.|........70997.|.259159...
k=5..|...5...|....41.|...31.|.1031.|..3881.|....19541.|.19531...
k=6..|..11..|....7...|...43.|.1549.|..9337.|....56131.|.55987...
...
k=10|..13..|..11...|1117|10111|101111|1111151|11110111|101111111|1111111121|.
========================================



>
>  "smallest base in which the n-th prime has 2 ones and zeros elsewhere"
>  I guess this is A006093(n).
>
>  These are all biased in even two ways: first there is the dependence on the
>  base, and second there is no reason to count the number of ones (as opposed to
>  2's, 3's,..).

In a sense, the only non-arbitrary digit is 0.

>
>  And there is some (not interesting IMO) auxiliary array of T(i,b) of
>  the number of ones of the i-th prime, i=1,2,3,.. in base b=2,3,4,...
>
>  1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
>  2,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
>  2,1,2,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
>  3,1,1,1,2,1,0,0,0,0,0,0,0,0,0,0,0,0,0,
>  3,1,0,1,1,1,1,1,2,1,0,0,0,0,0,0,0,0,0,
>  3,3,1,0,1,1,1,1,1,1,2,1,0,0,0,0,0,0,0,
>  2,1,2,0,0,0,1,1,1,1,1,1,1,1,2,1,0,0,0,
>  3,1,1,0,1,0,0,1,1,1,1,1,1,1,1,1,2,1,0,
>  4,1,2,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,
>  4,1,2,1,0,1,0,0,0,0,0,0,1,1,1,1,1,1,1,
>  5,3,1,3,1,0,0,0,1,0,0,0,0,1,1,1,1,1,1,
>  3,3,2,1,2,0,0,1,0,0,1,0,0,0,0,0,1,1,1,
>  3,3,1,2,1,0,1,0,1,0,0,0,0,0,0,0,0,0,1,
>  4,3,0,1,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,
>  5,1,0,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
>  4,1,2,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,
>  5,1,0,1,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,
>  5,1,1,1,2,2,0,0,1,0,1,0,0,1,0,0,0,0,1,
>  3,3,1,0,2,1,1,0,0,1,0,0,0,0,0,0,0,0,0,
>  4,1,2,1,1,2,1,0,1,0,0,0,1,0,0,0,0,0,0,
>
>  which contains A014499 in the first column, A073780 in the 2nd column.
>
>
>  Richard
>




This is just a note to let people know that
David Applegate and I are working on several
improvements to the OEIS.

The goal is to make it easier for people to send in comments,
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I'm also working on the backlog of existing comments
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The purpose of the "summer rules" is that I don't have
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Neil