A008336 -id:A008336 - cp's OEIS frontend

A370974 A260850 sorted into increasing order and duplicates omitted.

1, 2, 6, 20, 24, 120, 140, 858, 924, 1008, 1120, 1430, 10080, 11088, 12012, 22880, 176358, 388960, 1662804, 1750320, 3879876, 4056234, 9694845, 10029150, 10400600, 10816624, 33256080, 270415600, 280816200, 290845350, 300540195, 1037158320, 1452021648, 3181073742, 3267048708, 9617286240, 13784652882, 20583576819, 35263382880, 120880802196

Offset: 1

N. J. A. Sloane, Apr 15 2024

nonn

It seems very likely that there are no duplicates in A260850. Compare the proof of the analogous property of A008336.

Michael De Vlieger, Table of n, a(n) for n = 1..3365
Michael De Vlieger, Plot p(i)^m(i) | a(n) at (x,y) = (n,i), n = 1..2048, with a color function where black indicates m(i) = 1, red indicates m(i) = 2, ..., magenta indicates the largest m(i) for n <= 2048.

Cf. A008336, A065422, A260850, A370968.

A374317 For any n > 0, let b_n(n+1) = 1, and for k = 1..n, if k divides b_n(k+1) then b_n(k) = b_n(k+1) / k otherwise b_n(k) = b_n(k+1) * k; a(n) = b_n(1).

Original entry on oeis.org

1, 2, 6, 6, 30, 20, 140, 70, 70, 7, 77, 8316, 108108, 858, 5720, 1430, 24310, 48620, 923780, 46189, 3879876, 176358, 4056234, 2704156, 676039, 104006, 312018, 44574, 1292646, 4308820, 133573420, 267146840, 2203961430, 259289580, 363005412, 3267048708

Offset: 1

Rémy Sigrist, Jul 04 2024

nonn

This sequence is a variant of A008336; here we divide or multiply by numbers from n down to 1, there by numbers from 1 up to n.

Unlike A008336, this sequence contains several repeated terms.

			The first terms, alongside the corresponding sequences b_n, are:
  n   a(n)  b_n
  --  ----  ----------------------------------------------
   1     1  [1, 1]
   2     2  [2, 2, 1]
   3     6  [6, 6, 3, 1]
   4     6  [6, 6, 12, 4, 1]
   5    30  [30, 30, 60, 20, 5, 1]
   6    20  [20, 20, 40, 120, 30, 6, 1]
   7   140  [140, 140, 280, 840, 210, 42, 7, 1]
   8    70  [70, 70, 140, 420, 1680, 336, 56, 8, 1]
   9    70  [70, 70, 35, 105, 420, 84, 504, 72, 9, 1]
  10     7  [7, 7, 14, 42, 168, 840, 5040, 720, 90, 10, 1]

Rémy Sigrist, Table of n, a(n) for n = 1..1000
Rémy Sigrist, Colored representation of b_n(k) for n <= 1000 (where the color at (x, y) is function of b_x(y))

Cf. A008336, A374318.

a(n) = { my (b = 1); forstep (k = n, 1, -1, if (b % k==0, b /= k, b *= k);); return (b); }

A140644 a(0)=1. a(n) = a(n-1)/d(n) if d(n) divides a(n-1). Otherwise, a(n) = a(n-1)*d(n). (d(n) is the number of positive divisors of n.).

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 24, 12, 3, 1, 4, 2, 12, 6, 24, 6, 30, 15, 90, 45, 270, 1080, 270, 135, 1080, 360, 90, 360, 60, 30, 240, 120, 20, 5, 20, 5, 45, 90, 360, 90, 720, 360, 45, 90, 15, 90, 360, 180, 18, 6, 1, 4, 24, 12, 96, 24, 3, 12, 3, 6, 72, 36, 9, 54, 378, 1512, 189, 378, 63

Offset: 0

Leroy Quet, Jul 08 2008

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A008336.

A140644 := proc(n) option remember ; local d ; if n = 0 then 1; else d := numtheory[tau](n) ; if A140644(n-1) mod d = 0 then RETURN( A140644(n-1)/d ) ; else RETURN( d*A140644(n-1) ) ; fi; fi; end: for n from 0 to 80 do printf("%d,",A140644(n)) ; od: # R. J. Mathar, Aug 08 2008

nxt[{n_,a_}]:=Module[{d=DivisorSigma[0,n+1]},{n+1,If[Divisible[a,d], a/d, a*d]}]; Transpose[NestList[nxt,{0,1},70]][[2]] (* Harvey P. Dale, Jul 25 2015 *)

Extended beyond a(20) by R. J. Mathar, Aug 08 2008

A195458 a(n) = floor(sqrt(n)) * a(n-1), starting with 1.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 16, 32, 96, 288, 864, 2592, 7776, 23328, 69984, 279936, 1119744, 4478976, 17915904, 71663616, 286654464, 1146617856, 4586471424, 18345885696, 91729428480, 458647142400, 2293235712000, 11466178560000, 57330892800000, 286654464000000

Offset: 1

Peter Luschny, Sep 18 2011

nonn

a(n) = r(n+2)/sqrt(2); r(1) = sqrt(2); r(n) = r(n-1)/sqrt(n-1) if r(n-1) is a square else r(n) = r(n-1)*floor(sqrt(n-1).

A variation of Recamán's A008336.

Cf. A008336.

r := proc(n) option remember; if n = 1 then sqrt(2)
elif type(r(n-1),square) then r(n-1)/sqrt(n-1)
else r(n-1)*floor(sqrt(n-1)) fi end:
A195458 := proc(n) r(n+2)/sqrt(2) end:

a[1] = 1;
a[n_] := a[n] = Floor[Sqrt[n]] a[n - 1]
Table[a[n], {n, 20}] (* David Callan, Aug 14 2013 *)

a(n) = Product_{k=1..n} floor(sqrt(k)). - Ridouane Oudra, Feb 16 2023

Better name from David Callan, Aug 14 2013

A323615 a(0) = 1; for n > 0, a(n) = floor(a(n-1)/n) if positive and not already in the sequence, otherwise a(n) = a(n-1)*n.

Original entry on oeis.org

1, 1, 2, 6, 24, 4, 24, 3, 24, 216, 21, 231, 19, 247, 17, 255, 15, 255, 14, 266, 13, 273, 12, 276, 11, 275, 10, 270, 9, 261, 8, 248, 7, 231, 7854, 224, 8064, 217, 5, 195, 7800, 190, 7980, 185, 8140, 180, 8280, 176, 8448, 172, 8600, 168, 8736, 164

Offset: 0

Jan Koornstra, Jan 20 2019

Variation on A008336 using floor division and A076039 not allowing for values already in the sequence.

			a(5) = 4, since floor(24/5) = 4, which is positive and not already in the sequence.
a(6) = 24, since floor(4/6) = 0, hence not positive.

Jan Koornstra, Table of n, a(n) for n = 0..10000

Cf. A008336, A076039.

Nest[Append[#1, If[And[#3 > 0, FreeQ[#1, #3]], #3, #2 #1[[-1]] ]] & @@ {#1, #2, Floor[#1[[-1]]/#2]} & @@ {#, Length@ #} &, {1}, 53] (* Michael De Vlieger, Jan 23 2019 *)

def a323615(n):
  seq = []
  for i in range(n + 1):
    if i == 0: x = 1
    else:
      x = seq[i - 1] // i
      if x in seq or x == 0: x = seq[i - 1] * i
    seq.append(x)
  return seq
print(a323615(100))

A357839 a(n) is the greatest divisor > 1 of n which has already been listed, otherwise a(n) is the smallest number not yet listed; a(1) = 0.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 3, 2, 5, 4, 6, 2, 5, 4, 7, 6, 8, 5, 7, 2, 9, 8, 5, 2, 9, 7, 10, 10, 11, 8, 11, 2, 7, 9, 12, 2, 3, 10, 13, 7, 14, 11, 9, 2, 15, 12, 7, 10, 3, 13, 16, 9, 11, 14, 3, 2, 17, 15, 18, 2, 9, 16, 13, 11, 19, 17, 3, 14, 20, 18, 21, 2, 15, 19, 11

Offset: 1

Samuel Harkness, Oct 14 2022

When n is prime, a(n) is the prime index (A000720).

			For n = 6 the set of all divisors of 6 greater than 1 is {2, 3, 6}. Also, the set of all a(n < 6) is {0, 1, 2, 3}. The greatest divisor of 6 (excluding 1) that has been listed is 3, so a(6) = 3.

Samuel Harkness, Table of n, a(n) for n = 1..10000
Samuel Harkness, Log-log Scatterplot of the first 3000000 terms
Samuel Harkness, Scatterplot of the first 3000000 terms

Cf. A008336, A008344, A051352.

a = 0; A = {a}; Do[s = Drop[Reverse[Divisors[n]], 1]; s = Drop[s, -1]; If[Length[s] >= 1, Do[If[MemberQ[A, Part[s, d]], AppendTo[A, Part[s, d]]; Break[]], {d, 1, Length[s]}], a++; AppendTo[A, a]], {n, 2, 77}] Print[A]

first(n)=my(v=vector(n),m); forfactored(k=2,n, v[k[1]]=if(vecsum(k[2][,2])==1, m++, my(t); fordiv(k,d, if(d<=m, t=d)); t)); v \\ Charles R Greathouse IV, Oct 14 2022

A360300 a(n) is the least term in the n-th row of A360298.

Original entry on oeis.org

1, 2, 6, 24, 120, 20, 140, 630, 70, 7, 77, 924, 12012, 858, 5720, 12870, 218790, 12155, 230945, 46189, 969969, 176358, 4056234, 676039, 676039, 104006, 312018, 44574, 1292646, 1077205, 33393355, 66786710, 2203961430, 64822395, 90751353, 90751353, 3357800061

Offset: 1

Rémy Sigrist, Feb 02 2023

nonn

			For n = 9:
- the 9th row of A360298 is (10080, 70, 5670, 4480, 362880),
- so a(9) = 70.

Cf. A008336, A360298.

{ for (n = 1, 37, if (n==1, r = [1], r = setunion(select(v -> v%n==0, r)/n, r*n)); print1 (r[1]", ")) }

a(n) <= A008336(n+1).

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

A362698 For n > 1, if n appears in the sequence, a(n) = a(n-1) - n if nonnegative and not already in the sequence, otherwise a(n) = a(n-1) + 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, 114, 798, 6384, 57456, 574560, 6320160, 526680, 6846840, 489060, 32604, 521664, 8868288, 159629184, 8401536, 168030720, 3528645120, 160392960, 3689038080, 3689038056, 92225951400, 2397874736400, 64742617882800, 1812793300718400, 52571005720833600

Offset: 1

Kelvin Voskuijl, Jul 07 2023

nonn

			a(2) = 2, as a(1) = 1 and 1 times 2 is 2.
a(6) = 114, as a(3) = 6 = n, thus a(6) = a(5) - 6 =  114.
a(7) = 798, as a(6) = 114 and 7 times 114 is 798.

Michael De Vlieger, Table of n, a(n) for n = 1..1996
Michael De Vlieger, Log log scatterplot of log_10 a(n), n = 2..2^18.

Cf. A008336, A362332, A340612.

nn = 120; c[] := False; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; j = 2; Do[k = If[c[n], If[And[! c[#], # >= 0], #, j + n] &[j - n], If[Divisible[j, n], j/n, j n]]; Set[{a[n], c[k], j}, {k, True, k}], {n, 3, nn}]; Array[a, nn] (* _Michael De Vlieger, Aug 30 2023 *)

More terms from Michael De Vlieger, Aug 30 2023

cp's OEIS Frontend

A370974 A260850 sorted into increasing order and duplicates omitted.

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A374317 For any n > 0, let b_n(n+1) = 1, and for k = 1..n, if k divides b_n(k+1) then b_n(k) = b_n(k+1) / k otherwise b_n(k) = b_n(k+1) * k; a(n) = b_n(1).

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A140644 a(0)=1. a(n) = a(n-1)/d(n) if d(n) divides a(n-1). Otherwise, a(n) = a(n-1)*d(n). (d(n) is the number of positive divisors of n.).

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A195458 a(n) = floor(sqrt(n)) * a(n-1), starting with 1.

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A323615 a(0) = 1; for n > 0, a(n) = floor(a(n-1)/n) if positive and not already in the sequence, otherwise a(n) = a(n-1)*n.

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A357839 a(n) is the greatest divisor > 1 of n which has already been listed, otherwise a(n) is the smallest number not yet listed; a(1) = 0.

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A360300 a(n) is the least term in the n-th row of A360298.

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A362698 For n > 1, if n appears in the sequence, a(n) = a(n-1) - n if nonnegative and not already in the sequence, otherwise a(n) = a(n-1) + 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.

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A362698 For n > 1, if n appears in the sequence, a(n) = a(n-1) - n if nonnegative and not already in the sequence, otherwise a(n) = a(n-1) + 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.