A359727 Beattific 'primes': numbers n > 1 not equal to floor(k*m*phi) or floor(k*m*phi^2) for any smaller element k in this sequence and any positive integer m.

2, 4, 7, 8, 13, 14, 17, 23, 24, 28, 30, 39, 40, 43, 46, 49, 50, 53, 59, 65, 66, 70, 72, 75, 76, 81, 86, 88, 92, 96, 98, 107, 108, 114, 117, 118, 123, 127, 134, 140, 143, 144, 149, 150, 153, 156, 159, 160, 163, 166, 175, 176, 179, 182, 185, 191, 195

Offset

1,1

Comments

Given r = (1+sqrt(5))/2 and s = r^2, we sieve the set {2,3,4,...}, where each time we discover a new "prime" p, we sieve out the numbers floor(p*r), floor(2p*r), floor(3p*r), ... and floor(p*s), floor(2p*s), floor(3p*s), ... It appears that significantly more than half the terms are even.

Links

Table of n, a(n) for n=1..57.

Index entries for sequences generated by sieves

Example

The Beattific prime 2 causes us to sieve out 3, 6, 9, ... and 5, 10, ...; then the next Beattific prime, 4, doesn't cause us to throw out anything new; then the next Beattific prime is 7.

Mathematica

bp[limit_] := (*Find all the Beattific primes up to limit*)
  Module[{r = (1 + Sqrt[5])/2, sieve = ConstantArray[1, limit]},
   Do[If[sieve[[n]] == 1,
     sieve[[Table[Floor[k n r], {k, (limit + 1)/(n r)}]]] = 0;
     sieve[[Table[Floor[k n r r], {k, (limit + 1)/(n r r)}]]] = 0],
    {n, 2, limit}];
   Rest@Flatten@Position[sieve, 1]];

Prog

(PARI) h(n)=(n+sqrtint(5*n^2))\2
list(lim)=my(v=vectorsmall(lim\=1, i, 1), u=List()); for(n=2, #v, if(v[n]==0, next); listput(u, n); forstep(k=n, h(lim+1)-lim-1, n, v[h(k)]=0); forstep(k=n, 2*lim+1-h(lim+1), n, v[h(k)+k]=0)); Vec(u) \\ Charles R Greathouse IV, Jan 25 2023

Crossrefs

Cf. A001622, A104457, A000201, A001950.

Sequence in context: A320628 A260488 A190808 * A018614 A176745 A287208

Adjacent sequences: A359724 A359725 A359726 * A359728 A359729 A359730

Keyword

nonn

Author

James Propp, Jan 11 2023

Status

approved