A356867 -id:A356867 - cp's OEIS frontend

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

David James Sycamore, Sep 21 2022

nonn

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.).

			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).

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Fan style binary tree, n = 1..2^14, with a color function showing primes in red, proper prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, where magenta represents powerful numbers that are not prime powers.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, with the same color function as immediately above.
Index entries for sequences that are permutations of the natural numbers

Cf. A005940, A356867, A356886.

Cf. A053644, A053645.

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 *)

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

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

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

A364902 Let x, y be the greatest exponents of 2, 3 respectively such that 2^x, 3^y do not exceed n and let k_2, k_3 be n - 2^x, and n - 3^y respectively. Then for n such that k_2 = 0 or k_3 = 0, a(n) = n, else a(n) is the least novel number Min{pa(k_2), qa(k_3)}, where p, q are primes not equal to either 2 or 3.

Original entry on oeis.org

1, 2, 3, 4, 5, 10, 15, 8, 9, 7, 14, 20, 25, 35, 50, 16, 11, 22, 21, 28, 55, 70, 75, 40, 45, 49, 27, 13, 26, 33, 44, 32, 17, 34, 39, 52, 65, 98, 100, 56, 63, 77, 80, 121, 110, 105, 140, 112, 143, 154, 147, 196, 245, 135, 91, 130, 165, 220, 160, 85, 170, 195, 260, 64, 19, 38, 51, 68

Offset: 1

David James Sycamore and Michael De Vlieger Sep 21 2023

nonn

Motivated by the recursion D(2) known to reproduce A005940, this sequence uses a compound version based on a squarefree semiprime (6) rather than a prime, in which the terms are generated by a greedy algorithm related to the distances between n and the greatest powers of 2, and 3 not exceeding n. After a(9) = 9 each power of 2 or 3 is followed by the smallest prime not yet in the sequence. (e.g. 11 follows 16, 13 follows 27, etc).
There are no multiples of 6 in this sequence.
For k > 2, if a(i) = prime(k) = p and a(j) = p^2 then j-i is a term in A006899 (e.g. a(17) = 11, a(44) = 121 and 44 - 17 = 27 = 3^3).
Conjectures: (i). This is a permutation of A047253 with primes in order; (ii). All terms between consecutive prime terms, prime(k), prime(k+1) are prime(k)-smooth.

			a(n) = n for n <= 4 because all such n are powers of 2 or 3.
a(5) = least novel Min{a(1)*p,a(2)*q} = Min{p,2*q} for o,q prime != 2 or 3, so a(5) = 5.
17=16+1=9+8, so a(17) = least novel Min{a(1)*p,a(8)*q} = Min{p,8*q} = 11.
Data can be shown in tabular form in two distinct ways: First row starts with 1 and then rows start with a prime; alternatively each row starts with 2^i or 3^j:
 1;                       1;
 2;                       2;
 3,4;                     3;
 5,10,15,8,9;             4,5,10,15;
 7,14,20,25,35,50,16;     8;
 11,22,21,28,55...        9,7,14,20,25,35,50

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16.

Cf. A005940, A006899, A047253, A108906, A356867, A364611, A364628.

nn = 120; c[_] = False; s = {1, 2}; w = Length[s]; t = Prime[s]; flag = 0;
Array[Set[{q[#1], p[#1],
      r[#1]}, {#1, #2,
        Prepend[#2^Range[Floor@Log[#2, nn]], 1]} & @@ {#2,
       Prime[#2]}] & @@ {#, s[[#]]} &, w];
Do[If[n == 1,
   Set[{a[n], c[1]}, {1, True}],
   Array[Set[m[#], 1] &, w];
   Array[Set[j[#], n - p[#]^(-1 + LengthWhile[r[#], # < n + 1 &])] &, w];
   Array[
    If[j[#] == 0,
      k[#] = n; flag = #,
      While[Set[k[#], Prime[m[#]] a[j[#]]];
       Or[MemberQ[s, m[#]], c[k[#]]], m[#]++]] &, w];
   If[flag > 0,
    Set[{a[n], c[k[flag]]}, {k[flag], True}]; flag = 0,
    Set[{a[n], c[#]}, {#, True}] &[Min@ Array[k, w]] ]], {n, nn}];
Array[a, nn] (* Michael De Vlieger, Sep 24 2023 *)

For n > 6, a(A006899(n) + 1) = prime(n-2).

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A365389 a(n) = A365390(n) - 1.

Views

Author

Keywords

Links

Crossrefs