A065843 Let u be any string of n digits from {0,1}; let f(u) = number of distinct primes, not beginning with 0, formed by permuting the digits of u to a base-2 number; then a(n) = max_u f(u).

0, 1, 1, 2, 2, 3, 5, 12, 11, 24, 34, 79, 105, 194, 362, 734, 1143, 2045, 3872, 7758, 13001, 23902, 45539, 90436, 159510, 296210, 563833, 1110387, 2030754, 3876871, 7333827, 14353074, 26730538, 51246344, 97529176, 190928828, 358117285, 694240090, 1324674524, 2587693929, 4903604087, 9547001123

Offset

1,4

Links

Table of n, a(n) for n=1..42.

Example

a(4)=2 because 1011 and 1101 in base-2 notation are primes (11 and 13) and there is no set of three or more 4-digit primes with a common number of ones.

Maple

A065843 := proc(n)
    local b, u, udgs, uperm, a;
    b :=2 ;
    a := 0 ;
    for u from b^(n-1) to b^n-1 do
        udgs := convert(u, base, b) ;
        prs := {} ;
        for uperm in combinat[permute](udgs) do
            if op(-1, uperm) <> 0 then
                p := add( op(i, uperm)*b^(i-1), i=1..nops(uperm)) ;
                if isprime(p) then
                    prs := prs union {p} ;
                end if;
            end if;
        end do:
        a := max(a, nops(prs)) ;
    end do:
    a ;
end proc:
for n from 1 do
    print(n, A065843(n)) ;
end do: # R. J. Mathar, Apr 23 2016

Mathematica

c[x_] := Module[{},
   Length[Select[Permutations[x],
     First[#] != 0 && PrimeQ[FromDigits[#, 2]] &]]];
A065843[n_] := Module[{i},
   Return[Max[Map[c, DeleteDuplicatesBy[Tuples[Range[0, 1], n],
       Table[Count[#, i], {i, 0, 1}] &]]]]];
Table[A065843[n], {n, 1, 19}] (* Robert Price, Mar 30 2019 *)

Prog

(PARI) lista(n) = {my(m = matrix(n, n), c); forprime(i=2, 2^n, b = binary(i); m[#b, hammingweight(b)]++); vector(n, i, vecmax(m[i, ]))} \\ David A. Corneth, Apr 23 2016
(Python)
from sympy import isprime
from itertools import combinations_with_replacement as mc
from sympy.utilities.iterables import multiset_permutations as mp
def a(n): return n-1 if n < 3 else max(sum(1 for p in mp(c) if isprime(int("1"+"".join(p)+"1", 2))) for c in mc("01", n-2))
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Oct 09 2022

Crossrefs

Cf. A007053, A065844, A065845, A065846, A065847 A065848, A065849, A065850, A065851, A065852, A065853.

Sequence in context: A005426 A014644 A089541 * A111264 A065994 A152446

Adjacent sequences: A065840 A065841 A065842 * A065844 A065845 A065846

Keyword

base,nonn

Author

Sascha Kurz, Nov 24 2001

Extensions

6 more terms from Sean A. Irvine, Sep 06 2009

a(37)-a(39) from Michael S. Branicky, May 30 2024

a(40)-a(42) from Michael S. Branicky, Jun 14 2024

Status

approved