A008336 -id:A008336 - cp's OEIS frontend

A260850 Lexicographically earliest sequence such that for any n>1, n=u*v, where u/v = a(n)/a(n-1) in reduced form.

1, 2, 6, 24, 120, 20, 140, 1120, 10080, 1008, 11088, 924, 12012, 858, 1430, 22880, 388960, 1750320, 33256080, 1662804, 3879876, 176358, 4056234, 10816624, 270415600, 10400600, 280816200, 10029150, 290845350, 9694845, 300540195, 9617286240, 35263382880

Offset: 1

Paul Tek, Aug 01 2015

nonn

			From _Michael De Vlieger_, Apr 12 2024: (Start)
Table showing exponents m of prime powers p^m | a(n), n = 1..20, with "." representing p < gpf(n) does not divide a(n):
                       1111
    n        a(n)  23571379
   ------------------------
    1          1   .
    2          2   1
    3          6   11
    4         24   31
    5        120   311
    6         20   2.1
    7        140   2.11
    8       1120   5.11
    9      10080   5211
   10       1008   42.1
   11      11088   42.11
   12        924   21.11
   13      12012   21.111
   14        858   11..11
   15       1430   1.1.11
   16      22880   5.1.11
   17     388960   5.1.111
   18    1750320   421.111
   19   33256080   421.1111
   20    1662804   22..1111 (End)

Paul Tek, Table of n, a(n) for n = 1..3365
Paul Tek, PARI program for this sequence
Michael De Vlieger, Plot p(i)^m(i) | a(n) at (x,y) = (n,i), n = 1..2048, 3X vertical exaggeration, with a color function showing m(i) = 1 in black, m(i) = 2 in red, ..., largest m(i) in the dataset in magenta.
Michael De Vlieger, Prime Power Decomposition of a(n), n = 1..1000.

Cf. A008336, A370974 (sorted version).

nn = 35; p[_] := 0; r = 0;
Do[(Map[If[p[#1] < #2,
      p[#1] += #2,
      p[#1] -= #2] & @@ # &, #];
      If[r < #, r = #] &[#[[-1, 1]] ] ) &@
    Map[{PrimePi[#1], #2} & @@ # &, FactorInteger[n]];
  a[n] = Times @@ Array[Prime[#]^p[#] &, r], {n, nn}];
Array[a, nn] (* Michael De Vlieger, Apr 12 2024 *)

PARI
```
\\ See Links section.
```

a(p) = p*a(p-1) for any prime p.

a(n) = A008336(n+1) for n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 21, 22, 23; are there other indices with this property?

A065422 If n | a(n) then a(n+1) = a(n)/(highest power of n that divides a(n)), otherwise a(n+1) = n*a(n); a(0) = 1.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 20, 140, 1120, 10080, 1008, 11088, 77, 1001, 14014, 210210, 3363360, 57177120, 1029188160, 19554575040, 977728752, 2217072, 100776, 2317848, 96577, 2414425, 62775050, 1694926350, 47457937800, 1376280196200

Offset: 0

Naohiro Nomoto, Nov 23 2001

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Recamán's sequence

See A008336 for another version. Cf. A005132.

a065422 n = a065422_list !! n
a065422_list = 1 : 1 : f 2 1 where
   f n x = x' : f (n+1) x' where
       x' | x `mod` n == 0 = until ((> 0) . (`mod` n)) (`div` n) x
          | otherwise      = x * n
-- Reinhard Zumkeller, Oct 10 2011

nxt[{n_,a_}]:={n+1,If[Divisible[a,n],a/n^IntegerExponent[a,n],a*n]}; Join[ {1,1},Transpose[NestList[nxt,{3,2},30]][[2]]] (* Harvey P. Dale, Jan 08 2013 *)

Definition and offset corrected by Reinhard Zumkeller, Oct 10 2011

Typo in Crossrefs fixed by Paul Tek, Aug 01 2015

A371906 a(n) = sum of 2^(k-1) such that floor(n/prime(k)) is odd.

Original entry on oeis.org

0, 1, 3, 2, 6, 5, 13, 12, 14, 11, 27, 24, 56, 49, 55, 54, 118, 117, 245, 240, 250, 235, 491, 488, 492, 461, 463, 454, 966, 961, 1985, 1984, 2002, 1939, 1951, 1948, 3996, 3869, 3903, 3898, 7994, 7985, 16177, 16160, 16166, 15911, 32295, 32292, 32300, 32297, 32363

Offset: 1

Michael De Vlieger, Apr 15 2024

The only powers of 2 in the sequence are likely 1 and 2.

			a(1) = 0 since n = 1 is the empty product.
a(2) = 1 since for n = prime(1) = 2, floor(2/2) = 1 is odd. Therefore a(2) = 2^(1-1) = 1.
a(3) = 3 since for n = 3 and prime(1) = 2, floor(3/2) = 1 is odd, and for prime(2) = 3, floor(3/3) = 1 is odd. Hence a(3) = 2^(1-1) + 2^(2-1) = 1 + 2 = 3.
a(4) = 2 since for n = 4 and prime(1) = 2, floor(4/2) = 2 is even, but for prime(2) = 3, floor(4/3) = 1 is odd. Therefore, a(n) = 2^(2-1) = 2.
a(5) = 6 since for n = 5, though floor(5/2) = 2 is even, floor(5/3) and floor(5/5) are both odd. Therefore, a(n) = 2^(2-1) + 2^(3-1) = 2 + 4 = 6, etc.
Table relating a(n) with b(n), diagramming powers of 2 with "x" that sum to a(n), or prime factors with "x" that produce b(n), where b(n) = A372000(n).
             Power of 2
   n   a(n)  01234567      b(n)
  ----------------------------
   1     0   .               1
   2     1   x               2
   3     3   xx              6
   4     2   .x              3
   5     6   .xx            15
   6     5   x.x            10
   7    13   x.xx           70
   8    12   ..xx           35
   9    14   .xxx          105
  10    11   xx.x           42
  11    27   xx.xx         462
  12    24   ...xx          77
  13    56   ...xxx       1001
  14    49   x...xx        286
  15    55   xxx.xx       4290
  16    54   .xx.xx       2145
  17   118   .xx.xxx     36465
  18   117   x.x.xxx     24310
  19   245   x.x.xxxx   461890
  20   240   ....xxxx    46189
  ----------------------------
                 1111
             23571379
             Prime factor

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Plot powers 2^(i-1) that sum to a(n) at (x,y) = (n,i) for n = 1..2048.

Cf. A008336, A260850, A372000.

Table[Total[2^(-1 + Select[Range@ PrimePi[n], OddQ@ Quotient[n, Prime[#]] &])], {n, 50}]

a(n) = sum(k=1, primepi(n), if (n\prime(k) % 2, 2^(k-1))); \\ Michel Marcus, Apr 16 2024

A372000 a(n) = product of primes p such that floor(n/p) is odd.

Original entry on oeis.org

1, 2, 6, 3, 15, 10, 70, 35, 105, 42, 462, 77, 1001, 286, 4290, 2145, 36465, 24310, 461890, 46189, 969969, 176358, 4056234, 676039, 3380195, 520030, 1560090, 111435, 3231615, 430882, 13357342, 6678671, 220396143, 25928958, 907513530, 151252255, 5596333435, 589087730, 22974421470, 2297442147

Offset: 1

Michael De Vlieger, Apr 15 2024

The only primes in the sequence are 2 and 3.
We can approach the sequence in a manner akin to A260850, a variant of A008336. Set k = 1. Then for all prime factors p | n, if p | k, divide k by p, otherwise multiply k by p. Then we set a(n) = k. This accounts for the "toggling on or off" of prime factors as n increases.
For n >= 1, A055773(n) | a(n), where A055773(n) = A034386(n) / A034386(floor(n/2)).

			a(1) = 1 since n = 1 is the empty product.
a(2) = 2 since for n = 2, floor(n/p) = floor(2/2) = 1 is odd.
a(3) = 6 since for n = 3 and p = 2, floor(3/2) = 1 is odd, and for p = 3, floor(3/3) = 1 is odd. Hence a(3) = 2*3 = 6.
a(4) = 3 since for n = 4 and p = 2, floor(4/2) = 2 is even, but for p = 3, floor(4/3) = 1 is odd. Therefore, a(n) = 3.
a(5) = 15 since for n = 5, though floor(5/2) = 2 is even, floor(5/3) and floor(5/5) are both odd. Therefore, a(n) = 3*5 = 15, etc.
Table relating a(n) with b(n), diagramming prime factors with "x" that produce a(n), or powers of 2 with "x" that sum to b(n), where b(n) = A371906(n).
                Prime factor
                    1111
   n      b(n)  23571379   b(n)
  ----------------------------
   1        1   .            0
   2        2   x            1
   3        6   xx           3
   4        3   .x           2
   5       15   .xx          6
   6       10   x.x          5
   7       70   x.xx        13
   8       35   ..xx        12
   9      105   .xxx        14
  10       42   xx.x        11
  11      462   xx.xx       27
  12       77   ...xx       24
  13     1001   ...xxx      56
  14      286   x...xx      49
  15     4290   xxx.xx      55
  16     2145   .xx.xx      54
  17    36465   .xx.xxx    118
  18    24310   x.x.xxx    117
  19   461890   x.x.xxxx   245
  20    46189   ....xxxx   240
  ----------------------------
                01234567
                Power of 2

Michael De Vlieger, Table of n, a(n) for n = 1..3384
Michael De Vlieger, Plot prime(i) | a(n) at (x,y) = (n,i) for n = 1..2048.

Cf. A005117, A008336, A034386, A055773, A260850, A371906.

Table[Times @@ Select[Prime@ Range@ PrimePi[n], OddQ@ Quotient[n, #] &], {n, 40}] (* or *)
Table[Product[Prime[i], {j, 1 + Floor[PrimePi[n]/2]}, {i, 1 + PrimePi[Floor[n/(2 j)]], PrimePi[Floor[n/(2 j - 1)]]}], {n, 40}]

a(n) = vecprod(select(x->((n\x) % 2), primes([1, n]))); \\ Michel Marcus, Apr 16 2024

print([prod(p for p in prime_range(n + 1) if is_odd(n//p)) for n in range(1, 41)])
# Peter Luschny, Apr 16 2024

a(n) = Product_{k = 1..floor(pi(n)/2)+1} Product_{j = 1+floor(n/(2*k))..floor(n/(2*k-1))} prime(j), where pi(x) = A000720(n).

A326889 a(1) = 1; thereafter a(n) = a(n-1) / phi(n) if phi(n) divides a(n-1), otherwise a(n) = a(n-1) * phi(n), where phi is the Euler phi-function A000010.

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 12, 3, 18, 72, 720, 180, 15, 90, 720, 90, 1440, 240, 4320, 540, 45, 450, 9900, 79200, 3960, 330, 5940, 495, 13860, 110880, 3696, 231, 4620, 73920, 3080, 36960, 1330560, 73920, 3080, 49280, 1232, 14784, 352, 7040, 168960, 7680, 353280, 22080

Offset: 1

Rémy Sigrist, Sep 13 2019

nonn

This sequence has similarities with A008336 and with A008338.

			The first terms, alongside phi(n), are:
  n   a(n)  phi(n)
  --  ----  ------
   1     1       1
   2     1       1
   3     2       2
   4     1       2
   5     4       4
   6     2       2
   7    12       6
   8     3       4
   9    18       6
  10    72       4

Rémy Sigrist, Table of n, a(n) for n = 1..2500

See A327442 for an additive variant.

Cf. A000010, A008336, A008338, A336823.

for (n=1, 48, print1 (v=if (n==1, 1, v%e=eulerphi(n), v*e, v/e) ", "))

A330647 Lexicographically earliest sequence of distinct positive terms with an associate sequence t such that a(1) = t(1) = 1 and for n > 1, either a(n) divides t(n-1) (and in that case set t(n) = t(n-1)/a(n)) or a(n) is coprime to t(n-1) (and in that case set t(n) = t(n-1)*a(n)).

Original entry on oeis.org

1, 2, 3, 5, 6, 4, 7, 9, 10, 11, 13, 14, 8, 12, 17, 19, 22, 16, 23, 24, 15, 26, 28, 20, 21, 18, 25, 29, 30, 31, 32, 34, 37, 38, 40, 35, 41, 43, 44, 33, 27, 36, 47, 52, 39, 46, 51, 42, 48, 49, 53, 56, 58, 59, 61, 64, 60, 45, 62, 63, 57, 67, 68, 71, 73, 74, 77

Offset: 1

Rémy Sigrist, Dec 22 2019

nonn

All prime numbers appear in the sequence, in ascending order.

This sequence is likely a permutation of the natural numbers.

			The first terms, alongside the corresponding t(n), are:
  n   a(n)  t(n)
  --  ----  ----
   1     1     1
   2     2     2
   3     3     6
   4     5    30
   5     6     5
   6     4    20
   7     7   140
   8     9  1260
   9    10   126
  10    11  1386

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of (n, a(n)-n) for n = 1..250000
Rémy Sigrist, PARI program for A330647

See A330648 for the corresponding sequence t.

Cf. A008336.

Nest[Append[#1, Block[{k = 2, s}, While[Nand[FreeQ[#1[[All, 1]], k], MemberQ[{1, k}, Set[s, GCD[#3, k]]]], k++]; {k, If[s == 1, #3 k, #3/k]}]] & @@ {#, #[[-1, 1]], #[[-1, -1]]} &, {{1, 1}}, 66][[All, 1]] (* Michael De Vlieger, Dec 23 2019 *)

PARI
```
See Links section.
```

A360298 Irregular triangle (an infinite binary tree) read by rows. The tree has root node 1 in row n = 1. For n > 1, each node with value m in row n-1 has a left child with value m / n if n divides m, and a right child with value m * n.

Original entry on oeis.org

1, 2, 6, 24, 120, 20, 720, 140, 5040, 1120, 630, 40320, 10080, 70, 5670, 4480, 362880, 1008, 100800, 7, 700, 567, 56700, 448, 44800, 36288, 3628800, 11088, 1108800, 77, 7700, 6237, 623700, 4928, 492800, 399168, 39916800, 924, 133056, 92400, 13305600, 924, 92400, 74844, 51975, 7484400, 59136, 5913600, 33264, 4790016, 3326400, 479001600

Offset: 1

Rémy Sigrist, Feb 02 2023

This sequence is a variant of A360173; here we use divisions and multiplications, there subtractions and additions.

The n-th row has A360299(n) terms, starts with A008336(n+1) and ends with A000142(n).

			The tree begins:
  n     n-th row
  --    --------
   1    1___
            |
   2        2___
                |
   3            6___
                    |
   4               24___
                        |
   5      _____________120_____________
         |                             |
   6    20___                         720___
             |                              |
   7        140___                   _____5040_____
                  |                 |              |
   8            1120__           __630__       __40320__
                      |         |       |     |         |
   9                10080      70     5670  4480     362880

Rémy Sigrist, Table of n, a(n) for n = 1..11270 (rows for n = 1..26 flattened)

Cf. A000142, A008336, A360173, A360299 (row lengths), A360300.

row(n) = { my (r = [1]); for (h = 2, n, r=concat(apply(v -> if (v%h==0, [v/h, v*h], [v*h]), r))); return (r) }

from functools import cache
@cache
def row(n):
    if n == 1: return [1]
    out = []
    for r in row(n-1): out += ([r//n] if r%n == 0 else []) + [r*n]
    return out
print([an for r in range(1, 13) for an in row(r)]) # Michael S. Branicky, Feb 02 2023

T(n, 1) = A008336(n+1).
T(n, A360299(n)) = A000142(n).
T(p, k) = p * T(p-1, k) for any prime number p.

A362332 For n > 1, if n appears in the sequence then a(n) = lastindex(n), where lastindex(n) is the index of the last appearance of n. Otherwise a(n+1) = a(n)/(n+1) if (n+1)|a(n), otherwise a(n)(n+1), a(1) = 1 and a(2) = 12.

Original entry on oeis.org

1, 2, 6, 24, 120, 3, 21, 168, 1512, 15120, 166320, 13860, 180180, 12870, 858, 13728, 233376, 4200768, 79814592, 1596291840, 7, 154, 3542, 4, 100, 2600, 70200, 1965600, 57002400, 1900080, 58902480, 1884879360, 62201018880

Offset: 1

Kelvin Voskuijl, Apr 16 2023

nonn

			a(2) = 2, as a(1) = 1 and 1 times 2 is 2.
a(6) = 3, as a(3) = 6 = n, thus a(6) = 3.
a(7) = 21, as a(6) = 3 and 3 times 7 is 21.

Michael De Vlieger, Table of n, a(n) for n = 1..4306
Michael De Vlieger, Plot p(i)^e(i) | a(n) at (x,y) = (n,i) for n = 1..2048, with a color function representing e(i), where black represents e(i) = 1, red e(i) = 2, etc., 3X vertical exaggeration.

Cf. A008336, A340612.

nn = 33; c[] := 0; Array[Set[{a[#], c[#]}, {#, #}] &, 2]; j = 2; Do[If[c[n] > 0, Set[k, c[n]], If[Divisible[j, n], Set[k, j/n], Set[k, j*n]]]; Set[{a[n], c[k], j}, {k, n, k}], {n, 3, nn}]; Array[a, nn] (* _Michael De Vlieger, Jun 23 2023 *)

pos(n, va) = for (k=1, #va, if (va[k] == n, return (k)););
lista(nn) = my(va = vector(nn)); for (n=1, 2, va[n] = n); for (n=3, nn, my(p=pos(n, va)); if (p, va[n] = p, if (va[n-1] % n, va[n] = n*va[n-1], va[n] = va[n-1]/n); ); ); va; \\ Michel Marcus, Jun 23 2023

from itertools import count, islice
def A362332_gen(): # generator of terms
    a, ndict = 2, {1:1,2:2}
    yield from [1,2]
    for n in count(3):
        yield (a:= ndict[n] if n in ndict else (a*n if a%n else a//n))
        ndict[a] = n
A362332_list = list(islice(A362332_gen(),30)) # Chai Wah Wu, Jun 29 2023

A330648 a(1) = 1 and for any n > 1, if A330647(n) divides a(n-1) then a(n) = a(n-1) / A330647(n), otherwise a(n) = a(n-1) * A330647(n).

Original entry on oeis.org

1, 2, 6, 30, 5, 20, 140, 1260, 126, 1386, 18018, 1287, 10296, 858, 14586, 277134, 12597, 201552, 4635696, 193154, 2897310, 111435, 3120180, 156009, 7429, 133722, 3343050, 96948450, 3231615, 100180065, 3205762080, 94287120, 3488623440, 91805880, 2295147

Offset: 1

Rémy Sigrist, Dec 22 2019

nonn

This sequence has similarities with A008336.

			The first terms, alongside the corresponding A330647(n), are:
  n   a(n)  A330647(n)
  --  ----  ----------
   1     1           1
   2     2           2
   3     6           3
   4    30           5
   5     5           6
   6    20           4
   7   140           7
   8  1260           9
   9   126          10
  10  1386          11

Rémy Sigrist, Table of n, a(n) for n = 1..1000

Cf. A008336, A330647.

Nest[Append[#1, Block[{k = 2, s}, While[Nand[FreeQ[#1[[All, 1]], k], MemberQ[{1, k}, Set[s, GCD[#3, k]]]], k++]; {k, If[s == 1, #3 k, #3/k], If[Mod[#3, k] == 0, #3/k, #3 k]}]] & @@ {#, #[[-1, 1]], #[[-1, 2]], #[[-1, -1]]} &, {{1, 1, 1}}, 34][[All, -1]] (* Michael De Vlieger, Dec 23 2019 *)

x=1; s=0; for (n=1, 35, for (v=1, oo, if (!bittest(s,v), if (gcd(x,v)==1, s+=2^v; x*=v; break, x%v==0, s+=2^v; x/=v; break))); print1 (x", "))

A370969 First differences of A337486.

Original entry on oeis.org

5, 4, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 1, 1, 4, 2, 2, 4, 2, 6, 2, 2, 2, 6, 4, 2, 2, 4, 2, 1, 1, 1, 3, 1, 1, 4, 2, 4, 1, 3, 2, 2, 1, 3, 2, 2, 4, 2, 2, 1, 1, 6, 4, 1, 1, 2, 4, 2, 2, 3, 1, 2, 1, 1, 4, 1, 1, 3, 3, 2, 4, 2, 4, 4, 2, 2, 4, 2, 2, 4, 2, 2, 2, 6

Offset: 1

N. J. A. Sloane, Apr 11 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008336, A337486.

from itertools import count, islice
def A370969_gen(): # generator of terms
    c, m = 1, 1
    for n in count(2):
        a, b = divmod(c,n)
        if not b:
            yield n-m
            c, m = a, n
        else:
            c *= n
A370969_list = list(islice(A370969_gen(),30)) # Chai Wah Wu, Apr 11 2024

cp's OEIS Frontend

A260850 Lexicographically earliest sequence such that for any n>1, n=u*v, where u/v = a(n)/a(n-1) in reduced form.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A065422 If n | a(n) then a(n+1) = a(n)/(highest power of n that divides a(n)), otherwise a(n+1) = n*a(n); a(0) = 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

Extensions

A371906 a(n) = sum of 2^(k-1) such that floor(n/prime(k)) is odd.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A372000 a(n) = product of primes p such that floor(n/p) is odd.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

SageMath

Formula

A326889 a(1) = 1; thereafter a(n) = a(n-1) / phi(n) if phi(n) divides a(n-1), otherwise a(n) = a(n-1) * phi(n), where phi is the Euler phi-function A000010.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A330647 Lexicographically earliest sequence of distinct positive terms with an associate sequence t such that a(1) = t(1) = 1 and for n > 1, either a(n) divides t(n-1) (and in that case set t(n) = t(n-1)/a(n)) or a(n) is coprime to t(n-1) (and in that case set t(n) = t(n-1)*a(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A360298 Irregular triangle (an infinite binary tree) read by rows. The tree has root node 1 in row n = 1. For n > 1, each node with value m in row n-1 has a left child with value m / n if n divides m, and a right child with value m * n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

Formula

A362332 For n > 1, if n appears in the sequence then a(n) = lastindex(n), where lastindex(n) is the index of the last appearance of n. Otherwise a(n+1) = a(n)/(n+1) if (n+1)|a(n), otherwise a(n)*(n+1), a(1) = 1 and a(2) = 1*2.

A362332 For n > 1, if n appears in the sequence then a(n) = lastindex(n), where lastindex(n) is the index of the last appearance of n. Otherwise a(n+1) = a(n)/(n+1) if (n+1)|a(n), otherwise a(n)(n+1), a(1) = 1 and a(2) = 12.