A015999 -id:A015999 - cp's OEIS frontend

A033496 Numbers m that are the largest number in their Collatz (3x+1) trajectory.

1, 2, 4, 8, 16, 20, 24, 32, 40, 48, 52, 56, 64, 68, 72, 80, 84, 88, 96, 100, 104, 112, 116, 128, 132, 136, 144, 148, 152, 160, 168, 176, 180, 184, 192, 196, 200, 208, 212, 224, 228, 232, 240, 244, 256, 260, 264, 272, 276, 280, 288, 296, 304, 308, 312, 320, 324

Offset: 1

Jeff Burch

Or, possible peak values in 3x+1 trajectories: 1,2 and m=16k+4,16k+8,16k but not for all k; those 4k numbers [like m=16k+12 and others] which cannot be such peaks are listed in A087252.

Possible values of A025586(m) in increasing order. See A275109 (number of times each value of a(n) occurs in A025586). - Jaroslav Krizek, Jul 17 2016

			These peak values occur in 1, 3, 6, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 21, 22, 27, 30, 39, 44, 71, 75, 1579 [3x+1]-iteration trajectories started with different initial values. This list most probably is incomplete.
From _Hartmut F. W. Hoft_, Jun 24 2016: (Start)
Let n be the maximum in some Collatz trajectory and let F(n), the initial fan of n, be the set of all initial values less than or equal to n whose Collatz trajectories lead to n as their maximum. Then the size of F(n) never equals 2, 4, 5, 7 or 10 (see the link).
Conjecture: Every number k > 10 occurs as the size of F(n) for some n.
Fans F(n) of size k, for all 10 < k < 355, exist for 4 <= n <= 50,000,000. The largest fan in this range, F(41163712), has size 7450.
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 2000 terms from T. D. Noe)
Hartmut F. W. Hoft, initial Collatz fans
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A025586, A087251-A087256, A015999, A275109.
Cf. A006370, A008908, A070165.
Cf. A095384 (contains a definition of Collatz[]).
Cf. A105730, A233293.

a033496 n = a033496_list !! (n-1)
a033496_list = 1 : filter f [2, 4 ..] where
   f x = x == maximum (takeWhile (/= 1) $ iterate a006370 x)
-- Reinhard Zumkeller, Oct 22 2015

Set(Sort([Max([k eq 1 select n else IsOdd(Self(k-1)) and not IsOne(Self(k-1)) select 3*Self(k-1)+1 else Self(k-1) div 2: k in [1..5*n]]): n in [1..2^10] | Max([k eq 1 select n else IsOdd(Self(k-1)) and not IsOne(Self(k-1)) select 3*Self(k-1)+1 else Self(k-1) div 2: k in [1..5*n]]) le 2^10])) // Jaroslav Krizek, Jul 17 2016

Collatz[a0_Integer, maxits_:1000] := NestWhileList[If[EvenQ[ # ], #/2, 3# + 1] &, a0, Unequal[ #, 1, -1, -10, -34] &, 1, maxits]; (* Collatz[n] function definition by Eric Weisstein *)
Select[Range[324], Max[Collatz[#]] == # &] (* T. D. Noe, Feb 28 2013 *)

def a(n):
    if n<2: return [1]
    l=[n, ]
    while True:
        if n%2==0: n//=2
        else: n = 3*n + 1
        if n not in l:
            l.append(n)
            if n<2: break
        else: break
    return l
print([n for n in range(1, 501) if max(a(n)) == n]) # Indranil Ghosh, Apr 14 2017

A008908(a(n)) = A159999(a(n)). - Reinhard Zumkeller, May 04 2009

Max(A070165(a(n),k): k=1..A008908(a(n))) = A070165(a(n),1) = a(n). - Reinhard Zumkeller, Oct 22 2015

A016005 a(n) = (tau(n^9)+8)/9.

Original entry on oeis.org

1, 2, 2, 3, 2, 12, 2, 4, 3, 12, 2, 22, 2, 12, 12, 5, 2, 22, 2, 22, 12, 12, 2, 32, 3, 12, 4, 22, 2, 112, 2, 6, 12, 12, 12, 41, 2, 12, 12, 32, 2, 112, 2, 22, 22, 12, 2, 42, 3, 22, 12, 22, 2, 32, 12, 32, 12, 12, 2, 212, 2, 12, 22, 7, 12, 112, 2, 22, 12, 112, 2, 60, 2, 12

Offset: 1

Robert G. Wilson v

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A015996, A015999, A016001, A016002, A016003, A016006, A016007, A016008, A016009, A016012.

a[n_] := (DivisorSigma[0, n^9] + 8)/9; Array[a, 100] (* Amiram Eldar, Jan 27 2025 *)

A016005(n) = (numdiv(n^9)+8)/9; \\ Antti Karttunen, Jan 17 2017

Definition corrected by Vladeta Jovovic, Sep 03 2005

A015996 (tau(n^4) + 3)/4, where tau = A000005.

Original entry on oeis.org

1, 2, 2, 3, 2, 7, 2, 4, 3, 7, 2, 12, 2, 7, 7, 5, 2, 12, 2, 12, 7, 7, 2, 17, 3, 7, 4, 12, 2, 32, 2, 6, 7, 7, 7, 21, 2, 7, 7, 17, 2, 32, 2, 12, 12, 7, 2, 22, 3, 12, 7, 12, 2, 17, 7, 17, 7, 7, 2, 57, 2, 7, 12, 7, 7, 32, 2, 12, 7, 32, 2, 30, 2, 7, 12, 12, 7, 32, 2, 22, 5, 7, 2, 57, 7, 7

Offset: 1

Robert G. Wilson v

nonn

If n is prime, a(n) = 2 since a(p) = (tau(p^4)+3)/4 = (5+3)/4 = 2. - Wesley Ivan Hurt, Nov 16 2013

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A018892, A015995, A015999, A016001, A016002, A016003, A016005, A016006, A016007, A016008, A016009, A016012, A016020.

A015996 := proc(n)
        (numtheory[tau](n^4)+3)/4 ;
end proc; # R. J. Mathar, May 09 2013

Table[(DivisorSigma[0, n^4] + 3)/4, {n, 100}] (* Wesley Ivan Hurt, Nov 16 2013 *)

A015996(n) = (numdiv(n^4)+3)/4;
for(n=1, 10000, write("b015996.txt", n, " ", A015996(n)));
\\ Antti Karttunen, Jan 17 2017

a(n) = (A000005(n^4) + 3)/4.

Definition corrected by Vladeta Jovovic, Sep 03 2005

A016001 a(n) = (tau(n^6)+5)/6.

Original entry on oeis.org

1, 2, 2, 3, 2, 9, 2, 4, 3, 9, 2, 16, 2, 9, 9, 5, 2, 16, 2, 16, 9, 9, 2, 23, 3, 9, 4, 16, 2, 58, 2, 6, 9, 9, 9, 29, 2, 9, 9, 23, 2, 58, 2, 16, 16, 9, 2, 30, 3, 16, 9, 16, 2, 23, 9, 23, 9, 9, 2, 107, 2, 9, 16, 7, 9, 58, 2, 16, 9, 58, 2, 42, 2, 9, 16, 16, 9, 58, 2, 30, 5, 9, 2, 107, 9

Offset: 1

Robert G. Wilson v

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A015996, A015999, A016002, A016003, A016005, A016006, A016007, A016008, A016009, A016012.

a[n_] := (DivisorSigma[0, n^6] + 5)/6; Array[a, 100] (* Amiram Eldar, Jan 27 2025 *)

A016001(n) = (numdiv(n^6)+5)/6;
for(n=1, 10000, write("b016001.txt", n, " ", A016001(n)));
\\ Antti Karttunen, Jan 17 2017

Definition corrected by Vladeta Jovovic, Sep 03 2005

A016002 a(n) = (tau(n^7)+6)/7.

Original entry on oeis.org

1, 2, 2, 3, 2, 10, 2, 4, 3, 10, 2, 18, 2, 10, 10, 5, 2, 18, 2, 18, 10, 10, 2, 26, 3, 10, 4, 18, 2, 74, 2, 6, 10, 10, 10, 33, 2, 10, 10, 26, 2, 74, 2, 18, 18, 10, 2, 34, 3, 18, 10, 18, 2, 26, 10, 26, 10, 10, 2, 138, 2, 10, 18, 7, 10, 74, 2, 18, 10, 74, 2, 48, 2, 10, 18, 18

Offset: 1

Robert G. Wilson v

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A015996, A015999, A016001, A016003, A016005-A016009, A016012.

(DivisorSigma[0,Range[80]^7]+6)/7 (* Harvey P. Dale, Jul 27 2012 *)

A016002(n) = (numdiv(n^7)+6)/7;
for(n=1, 10000, write("b016002.txt", n, " ", A016002(n)));
\\ Antti Karttunen, Jan 17 2017

Definition corrected by Vladeta Jovovic, Sep 03 2005

A016003 a(n) = (tau(n^8)+7)/8.

Original entry on oeis.org

1, 2, 2, 3, 2, 11, 2, 4, 3, 11, 2, 20, 2, 11, 11, 5, 2, 20, 2, 20, 11, 11, 2, 29, 3, 11, 4, 20, 2, 92, 2, 6, 11, 11, 11, 37, 2, 11, 11, 29, 2, 92, 2, 20, 20, 11, 2, 38, 3, 20, 11, 20, 2, 29, 11, 29, 11, 11, 2, 173, 2, 11, 20, 7, 11, 92, 2, 20, 11, 92, 2, 54, 2, 11, 20, 20

Offset: 1

Robert G. Wilson v

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A015996, A015999, A016001, A016002, A016005-A016009, A016012.

Table[(DivisorSigma[0,n^8]+7)/8,{n,80}] (* Harvey P. Dale, Sep 22 2020 *)

A016003(n) = (numdiv(n^8)+7)/8;
for(n=1, 10000, write("b016003.txt", n, " ", A016003(n)));
\\ Antti Karttunen, Jan 17 2017

Definition corrected by Vladeta Jovovic, Sep 03 2005

A016009 a(n) = (tau(n^13)+12)/13.

Original entry on oeis.org

1, 2, 2, 3, 2, 16, 2, 4, 3, 16, 2, 30, 2, 16, 16, 5, 2, 30, 2, 30, 16, 16, 2, 44, 3, 16, 4, 30, 2, 212, 2, 6, 16, 16, 16, 57, 2, 16, 16, 44, 2, 212, 2, 30, 30, 16, 2, 58, 3, 30, 16, 30, 2, 44, 16, 44, 16, 16, 2, 408, 2, 16, 30, 7, 16, 212, 2, 30, 16, 212, 2, 84, 2, 16

Offset: 1

Robert G. Wilson v

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A015996, A015999, A016001, A016002, A016003, A016005, A016006, A016007, A016008, A016012.

a[n_] := (DivisorSigma[0, n^13] + 12)/13; Array[a, 100] (* Amiram Eldar, Jan 27 2025 *)

A016009(n) = (numdiv(n^13)+12)/13;
for(n=1, 10000, write("b016009.txt", n, " ", A016009(n)));
\\ Antti Karttunen, Jan 17 2017

Definition corrected by Vladeta Jovovic, Sep 03 2005

A016012 a(n) = (tau(n^n)+n-1)/n.

Original entry on oeis.org

1, 2, 2, 3, 2, 9, 2, 4, 3, 13, 2, 28, 2, 17, 18, 5, 2, 40, 2, 44, 24, 25, 2, 77, 3, 29, 4, 60, 2, 994, 2, 6, 36, 37, 38, 149, 2, 41, 42, 125, 2, 1894, 2, 92, 94, 49, 2, 198, 3, 104, 54, 108, 2, 167, 58, 173, 60, 61, 2, 7505, 2, 65, 130, 7, 68, 4558, 2, 140, 72, 5114

Offset: 1

Robert G. Wilson v

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A015996, A015999, A016001, A016002, A016003, A016005, A016006, A016007, A016008, A016009, A016025.

f[j_, n_] := (DivisorSigma[0, n^j] + j - 1)/j; Table[ f[n, n], {n, 96}]

A016012(n) = (numdiv(n^n)+n-1)/n;
for(n=1, 10000, write("b016012.txt", n, " ", A016012(n)));
\\ Antti Karttunen, Jan 17 2017

Definition corrected by Vladeta Jovovic, Sep 03 2005

A126098 Where records occur in A018892.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 30, 60, 120, 180, 210, 360, 420, 840, 1260, 1680, 2520, 4620, 7560, 9240, 13860, 18480, 27720, 55440, 83160, 110880, 120120, 180180, 240240, 360360, 720720, 1081080, 1441440, 1801800, 2042040, 2882880, 3063060, 4084080, 5405400, 6126120, 12252240, 18378360, 24504480

Offset: 1

N. J. A. Sloane, Mar 05 2007

nonn

Remarkably similar to but ultimately different from A018894. - Jorg Brown and N. J. A. Sloane, Mar 06 2007
This sequence represents "where records occur" for a number of sequences in addition to A018892 including the following: A015995, A015996, A015999, A016001, A016002, A016003, A016005, A016006, A016007, A016008, A016009, A048691, A048785, A063647, A117677, A144943. - Ray Chandler, Dec 04 2008
Subsequence of A025487. - Ray Chandler, Sep 05 2008
Also record-setting elements of tau(n^2) (just as A002182 gives the record-setting elements of tau(n)). The point is that A018892 is (tau(n^2) + 1)/2. As tau(n^2) is odd, the record-setting elements of A018892 are also the record setting elements of tau(n^2). - Allen Tracht, Jan 20 2009

Ray Chandler, Table of n, a(n) for n = 1..582
Jorg Brown, Comparison of records in sigma(n)/phi(n) and A018892

Cf. A018892, A126097. Equals A117010(n) + 1.

More terms from Jorg Brown (jorg(AT)google.com) and T. D. Noe, Mar 05 2007

a(27) corrected by hupo001(AT)gmail.com, Jan 10 2008

A016006 a(n) = (tau(n^10)+9)/10.

Original entry on oeis.org

1, 2, 2, 3, 2, 13, 2, 4, 3, 13, 2, 24, 2, 13, 13, 5, 2, 24, 2, 24, 13, 13, 2, 35, 3, 13, 4, 24, 2, 134, 2, 6, 13, 13, 13, 45, 2, 13, 13, 35, 2, 134, 2, 24, 24, 13, 2, 46, 3, 24, 13, 24, 2, 35, 13, 35, 13, 13, 2, 255, 2, 13, 24, 7, 13, 134, 2, 24, 13, 134, 2, 66, 2, 13

Offset: 1

Robert G. Wilson v

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A015996, A015999, A016001, A016002, A016003, A016005, A016007, A016008, A016009, A016012.

a[n_] := (DivisorSigma[0, n^10] + 9)/10; Array[a, 100] (* Amiram Eldar, Jan 27 2025 *)

A016006(n) = (numdiv(n^10)+9)/10;
for(n=1, 10000, write("b016006.txt", n, " ", A016006(n)));
\\ Antti Karttunen, Jan 17 2017

Definition corrected by Vladeta Jovovic, Sep 03 2005

cp's OEIS Frontend

A033496 Numbers m that are the largest number in their Collatz (3x+1) trajectory.

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A016005 a(n) = (tau(n^9)+8)/9.

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A015996 (tau(n^4) + 3)/4, where tau = A000005.

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A016001 a(n) = (tau(n^6)+5)/6.

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A016002 a(n) = (tau(n^7)+6)/7.

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A016003 a(n) = (tau(n^8)+7)/8.

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A016009 a(n) = (tau(n^13)+12)/13.

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A016012 a(n) = (tau(n^n)+n-1)/n.

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