A357268 If n is a power of 2, a(n) = n. Otherwise, if 2^j is the greatest power of 2 not exceeding n, and if k = n - 2^j, then a(n) is the smallest m*a(k) which has not occurred already, where m is an odd number.

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 15, 12, 25, 18, 27, 16, 11, 14, 21, 20, 35, 30, 45, 24, 49, 50, 75, 36, 125, 54, 81, 32, 13, 22, 33, 28, 55, 42, 63, 40, 77, 70, 105, 60, 175, 90, 135, 48, 99, 98, 147, 100, 245, 150, 225, 72, 343, 250, 375, 108, 625, 162, 243, 64, 17

Offset

1,2

Comments

The definition implies that if n is not a power of 2, then neither is a(n).

Similar to the Doudna sequence (A005940), except that here the multiple of a(k) used to compute a(n) is the least odd number (rather than the least odd prime), such that a(n) is a novel term. Terms are the same as in A005940 until a(49)=99 (instead of 121), subsequent to which further odd nonprime multiples produce more differences from A005940; the next is a(71)=117 (instead of 99).

A permutation of the positive integers, in which the primes appear in natural order, but the odd numbers do not (9 precedes 7, 25 precedes 21, etc.).

Links

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

Index entries for sequences that are permutations of the natural numbers

Formula

a(2^n + 1) is the smallest odd number which has not already occurred.

Example

n = 49 = 2^5 + 17, and a(17) = 11, so a(49) is the least m*a(17) which has not occurred earlier, where m is an odd number. Up to this point we have seen 3*11, 5*11, 7*11, but not 9*11. Therefore a(49) = 9*11 = 99 (compare with A005940(71)=99).

Mathematica

nn = 65; m = 1; c[_] = False; Do[Set[{m, k}, {1, n - 2^Floor[Log2[n]]}]; If[k == 0, Set[{a[n], c[n]}, {n, True}], While[Set[t, m a[k]]; Or[m == 1, c[t]], m += 2]; Set[{a[n], c[t]}, {t, True}]], {n, nn}]; Array[a, nn] (* Michael De Vlieger, Sep 21 2022 *)

Prog

(Python)
from sympy import nextprime
from sympy.ntheory import digits
from itertools import count, islice
def b(n): return n - 2**(len(bin(n)[2:]) - 1)
def agen():
    aset, alst = set(), [None]
    for n in count(1):
        k = b(n)
        if k == 0: an = n
        else:
            ak, p = alst[k], 3
            while p*ak in aset: p += 2
            an = p*ak
        yield an; aset.add(an); alst.append(an)
print(list(islice(agen(), 65))) # Michael S. Branicky, Sep 21 2022
(PARI) f(n) = n - 2^(logint(n, 2)); \\ A053645
lista(nn) = {my(va = vector(nn), sa = Set(va)); for (n=1, nn, my(x = f(n)); if (x == 0, va[n] = n, my(k=1); while (setsearch(sa, k*va[x]), k+=2); va[n] = k*va[x]; ); sa = Set(va); ); va; } \\ Michel Marcus, Sep 27 2022

Crossrefs

Cf. A005940, A356867, A356886.

Cf. A053644, A053645.

Sequence in context: A340364 A324106 A252753 * A005940 A332815 A355405

Adjacent sequences: A357265 A357266 A357267 * A357269 A357270 A357271

Keyword

nonn

Author

David James Sycamore, Sep 21 2022

Status

approved