A356886 Write n as 2^m - k, where 2^m is the least power of 2 such that 2^m >= n, and k is a number in the range 0 <= k < 2^(m-1) - 1. Then for n such that k=0, a(n)=n, and for n such that k > 0, a(n) is the smallest odd prime multiple of a(k) that is not already a term.

1, 2, 3, 4, 9, 6, 5, 8, 15, 18, 27, 12, 21, 10, 7, 16, 35, 30, 63, 36, 81, 54, 45, 24, 25, 42, 99, 20, 33, 14, 11, 32, 55, 70, 165, 60, 297, 126, 75, 72, 135, 162, 243, 108, 189, 90, 105, 48, 49, 50, 147, 84, 351, 198, 195, 40, 65, 66, 117, 28, 39, 22, 13, 64, 91

Offset

1,2

Comments

Conjectured to be a permutation of the positive integers in which the primes appear in order. The even bisection, when divided by 2 reproduces the sequence. Has similar properties to the Doudna sequence, A005940.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..16384

Michael De Vlieger, Annotated fan-style binary tree of a(n), n = 1..2^14, with row m = 2^m..2^(m+1)-1 with a heat map color function showing row minima in blue, larger terms in greens, and row maxima in red.

Formula

a(2^m - 1) = prime(m) for m >= 2.

a(2*n)/2 = a(n) for n >= 1.

Example

5 = 2^3 - 3 so a(5)=a(3)*3=9.
13 = 2^4 - 3 and a(3)=3 so a(13)=3*7=21 since 9 and 15 have appeared already.
17 = 2^5 - 15 and a(15)=7 so a(17)=5*7=35 (since 21=3*7 has appeared already).

Mathematica

nn = 65; c[_] = False; Do[Set[{m, k}, {2, 2^(Ceiling[Log2[n]]) - n}]; If[k == 0, Set[{a[n], c[n]}, {n, True}], While[Set[t, Prime[m] a[k]]; c[t], m++]; Set[{a[n], c[t]}, {t, True}]], {n, nn}]; Array[a, nn] (* Michael De Vlieger, Sep 02 2022 *)

Prog

(PARI) first(n) = { my(res = vector(n), m = Map()); for(i = 1, n, qd = ceil(log(i)/log(2)); nextp = 1<<qd; if(nextp == i, res[i] = i , k = res[nextp - i]; forprime(p = 3, oo, if(!mapisdefined(m, k*p), res[i] = k*p; mapput(m, k*p, 1); next(2) ) ) ) ); res } \\ David A. Corneth and Michel Marcus, Sep 13 2022
(Python)
from itertools import count, islice
from sympy import nextprime
def A356886_gen(): # generator of terms
    aset, alist = {1}, [1]
    yield 1
    for n in count(2):
        if k:=(0 if (i:=1<<n.bit_length())==n<<1 else i-n):
            p, m = 3, alist[k-1]
            while p*m in aset:
                p = nextprime(p)
            r = p*m
        else:
            r = n
        alist.append(r)
        aset.add(r)
        yield r
A356886_list = list(islice(A356886_gen(), 30)) # Chai Wah Wu, Sep 09 2022

Crossrefs

Cf. A005940.

Sequence in context: A106443 A091205 A106447 * A359416 A222248 A352728

Adjacent sequences: A356883 A356884 A356885 * A356887 A356888 A356889

Keyword

nonn

Author

David James Sycamore, Sep 02 2022

Extensions

More terms from David A. Corneth, Sep 02 2022

Status

approved