A356886 -id:A356886 - cp's OEIS frontend

A357057 a(n) = A356886(2^n+1)/A356886(2^n-1).

3, 3, 3, 5, 5, 7, 11, 11, 13, 17, 19, 19, 23, 29, 31, 37, 37, 41, 43, 47, 53, 59, 59, 61, 67, 71, 73, 79, 83, 83, 89

Offset: 1

Paul Curtz, Sep 09 2022

All terms are odd primes; some of them are repeated.

Conjecture: This sequence has the pattern: a prime repeated, a run of m times primes standing alone, the next prime repeated, a run of m+1 times primes standing alone, ... . All primes will be repeated once or follow in sequence. We know that if A356886(2^n-1) = p1 then A356886(2^n+1) will be p1*p2. p2 will be the smallest possible prime such that p1*p2 is not yet in the sequence A356886, thus p2 = a(n). Let p1*pn be already in A356886 and pn < p2, then we know that p1*pn will be on a position A356886(2^n-(2^(k+1)-2)) with some k > 0. This should explain this pattern. - Thomas Scheuerle, Sep 14 2022

Cf. A000051, A000225, A356886.

Cf. also A065091 (odd primes).

a(14)-a(24) from Michel Marcus, Sep 13 2022

a(25)-a(31) from Chai Wah Wu, Oct 01 2022

A365436 a(2^k) = 2^k for all k >= 0. let 2^r be the smallest power of 2 which exceeds n, then a(n) = the least novel m*a(k), where k = 2^r-n, and m is not a prior term.

Original entry on oeis.org

1, 2, 3, 4, 15, 10, 5, 8, 30, 60, 90, 24, 18, 12, 6, 16, 42, 84, 126, 168, 630, 420, 210, 56, 35, 70, 105, 28, 21, 14, 7, 32, 63, 154, 189, 252, 945, 770, 315, 504, 1890, 3780, 5670, 1512, 1134, 756, 378, 144, 54, 108, 162, 216, 810, 540, 270, 72, 45, 110, 135

Offset: 1

David James Sycamore, Nov 09 2023

nonn

Based on a recursion similar to that which produces the Doudna sequence, A005940, (using the least power of 2 exceeding n rather than the greatest power of 2 not exceeding n). All 2^(n-1) terms between between fixed points 2^n and 2^(n+1) are multiples m*a(k) of m, the least unused term, and m is a(2^(k+1)-1).
Conjectured to be a permutation of the positive integers.
From David A. Corneth, Nov 11 2023: (Start)
This is a permutation of the positive integers.
To prove this we'll show that each integer occurs at most once and at least once hence exactly once.
By definition (...a(n) = the least novel...) each positive integer occurs at most once.
Now suppose t is the smallest term not in the sequence. Then there exists u such that a(1)..a(u) contain the positive integers from 1 through t-1. Then a(i) = t for some 1 <= i <= 2^e - 1 where 2^e - 1 >= u. If a(i) != t for 1 <= i <= 2^e-2 then a(2^e - 1) = t as then k = 1, a(k) = 1 and m is not a prior term (t did not occur earlier).
Hence t occurs at least once. As it also occurs at most once every positive integer occurs exactly once and this sequence is a permutation of the positive integers. (End)

			a(3) = 3 since k = 1, a(1) = 1  and 3 is the smallest number which is not already a term.
a(5) = 15 since k = 8-5 = 3, a(3) = 3 and 5 is the smallest number which is not already a term.
a(31) = 7, the least unused term at this point in the sequence.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
David A. Corneth, PARI program.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^12, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue.
Michael De Vlieger, Fan style binary tree of a(n), n = 1..8191, showing primes in red, squares of primes in orange, other composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue, purple (also in A286708), and pink (also in A303606).
Index entries for sequences that are permutations of the natural numbers

Cf. A005940, A356886.

nn = 120;  c[] := False; c[1] = True; m[] := 1; a[1] = 1; c[1] = True;
 Do[If[IntegerQ[#],
      Set[k, i],
       While[Or[c[m[#]], c[Set[k, # m[#]]]], m[#]++] &[
         a[2^Floor[# + 1] - i]]] &@ Log2[i];
      Set[{a[i], c[k]}, {k, True}], {i, nn}];
  Array[a, nn] (* Michael De Vlieger, Nov 13 2023 *)

PARI
```
\\ See PARI link
```

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.

Original entry on oeis.org

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.

A359416 Write n as 2^m - k, where 2^m is the least power of 2 >= n (0 <= k <= 2^(m-1)-1). For n a power of 2 (k = 0), a(n) = n. For numbers with k > 0, a(n) is the least p*a(k) which has not occurred previously, the count of k being taken from right to left (backwards) from k = 1 at 2^m - 1.

Original entry on oeis.org

1, 2, 3, 4, 9, 6, 5, 8, 25, 18, 27, 12, 15, 10, 7, 16, 49, 50, 105, 36, 81, 54, 75, 24, 35, 30, 45, 20, 21, 14, 11, 32, 121, 98, 231, 100, 495, 210, 175, 72, 225, 162, 243, 108, 315, 150, 147, 48, 77, 70, 165, 60, 135, 90, 125, 40, 55, 42, 63, 28, 33, 22, 13, 64

Offset: 1

David James Sycamore, Dec 30 2022

nonn

A variant of the recursive definition of the Doudna sequence A005940, and A356886. Whereas a sequence is normally computed in natural order (A000027) of its indices (a(1), a(2), a(3), etc.), in this case terms with indices n other than powers of 2 are computed backwards, right to left from the least power of 2 exceeding n (ordering as in A122155). For example, with terms between a(4) and a(8) the order of computation is a(7), then a(6), then a(5), each time choosing a least novel number matching the definition, see Example. Compare with similar sequence A356886, where a similar definition is used but the count is conventional: left to right.

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

			a(3) = 3 because 3 = 2^2 - 1, so k = 1 and 3 is the least odd prime multiple of a(1).
a(7) = 5 because 7 = 2^3 - 1, k = 1, a(1) = 1 and 5 is the least multiple of 1 not seen already. (At this point a(5), a(6) have not been found.)
a(6) = 6 since k = 2, a(2) = 2, and 3*2 is the least number not seen already.
a(5) = 9 since k = 3, a(3) = 3, so we choose 3*3.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^12.
Michael De Vlieger, Labeled fan-style binary tree showing a(n), n = 1..2^12 in levels k such that n = 2^k+1..2^(k+1), with a color function indicating a(2^k) and a(n), where a(n) = a(2^k) in green, a(n) < a(2^k) in blues, and a(n) > a(2^k) in yellows, oranges, and reds.

Cf. A000027, A005940, A122155, A356886, A357057.

nn = 2^6; c[] = False; Do[Set[{m, k}, {2, n - 2^Floor[Log2[n]]}]; If[k == 0, Set[{a[n], c[n]}, {n, True}], While[Set[t, Prime[m] a[k]]; c[t], m++]; Set[{a[2^#/(1 + Boole[IntegerQ[Log2[n]]]) - n + 2^(# - 1) &@ IntegerLength[n, 2]], c[t]}, {t, True}]], {n, 2^Ceiling[Log2[nn]] }]; Array[a, nn] (* _Michael De Vlieger, Jan 02 2023 *)

a(2*n)/2 = a(n); n >= 1.
a(2^n - 1) = prime(n); n >= 2.
a(2^n + 1) = prime(n)^2; n >= 2.
At the occurrence of 2^n (n >= 3) the following pattern of five successive terms is observed: 3*prime(n-1), 2*prime(n-1), prime(n), 2^n, prime(n)^2, ....
For n >= 2, a(2^n + 1)/a(2^n - 1) = prime(n); compare with A357057).

cp's OEIS Frontend

A357057 a(n) = A356886(2^n+1)/A356886(2^n-1).

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A365436 a(2^k) = 2^k for all k >= 0. let 2^r be the smallest power of 2 which exceeds n, then a(n) = the least novel m*a(k), where k = 2^r-n, and m is not a prior term.

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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.

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A359416 Write n as 2^m - k, where 2^m is the least power of 2 >= n (0 <= k <= 2^(m-1)-1). For n a power of 2 (k = 0), a(n) = n. For numbers with k > 0, a(n) is the least p*a(k) which has not occurred previously, the count of k being taken from right to left (backwards) from k = 1 at 2^m - 1.

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