A061984 -id:A061984 - cp's OEIS frontend

A061987 Number of times n-th distinct value is repeated in sequence b(k) = 1 + b(floor(k/2)) + b(floor(k/3)) with b(0) = 0, i.e., in A061984; also number of times n-th distinct row is repeated in square array T(n,k) = T(n-1,k) + T(n-1,floor(k/2)) + T(n-1,floor(k/3)) with T(0,0) = 1, i.e., in A061980.

1, 1, 1, 1, 2, 2, 1, 3, 4, 2, 6, 3, 5, 4, 12, 6, 10, 8, 9, 15, 12, 20, 16, 18, 30, 24, 27, 13, 32, 36, 60, 48, 54, 26, 64, 72, 81, 39, 96, 108, 52, 128, 144, 162, 78, 192, 216, 104, 139, 117, 288, 324, 156, 384, 432, 208, 278, 234, 576, 648, 312, 417, 351, 864, 416, 556

Offset: 0

Henry Bottomley, May 24 2001

nonn

For n > 0: a(n) = A003586(n+1) - A003586(n) and a(A084791(n)) = A084788(n).
Also number of times A160519(n+1) is repeated in A088468. - Reinhard Zumkeller, May 16 2009
In the 14th century Levi Ben Gerson proved that a(n) > 1 for all n > 6; see A003586, A235365, A235366, A236210. - Jonathan Sondow, Jan 20 2014

Reinhard Zumkeller, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, Smooth Number

import Data.List (group)
a061987 n = a061987_list !! n
a061987_list = map length $ group a061984_list
-- Reinhard Zumkeller, Jan 11 2014

a(n) = A061986(A061985(n)).

More terms from Reinhard Zumkeller, Jun 03 2003

A061985 Values which can occur in sequence b(k) = 1 + b(floor(k/2)) + b(floor(k/3)) with b(0) = 0, i.e., in A061984.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 11, 12, 15, 19, 20, 21, 27, 32, 36, 37, 47, 48, 54, 64, 65, 80, 85, 92, 112, 113, 114, 135, 150, 158, 193, 199, 200, 228, 263, 264, 273, 329, 350, 351, 387, 457, 464, 474, 558, 614, 615, 616, 661, 787, 815, 826, 946, 1072, 1073, 1081, 1136

Offset: 0

Henry Bottomley, May 24 2001

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..100

Cf. A061984, A061986, A061987.

a061985 n = a061985_list !! n
a061985_list = f (-1) a061984_list where
   f u (v:vs) = if v == u then f u vs else v : f v vs
-- Reinhard Zumkeller, Jan 11 2014

a(n) = a(n-1) + C(A022328(n) + A022329(n), A022328(n)). - David Wasserman, Nov 17 2005

A061986 Number of times n appears in sequence b(k) = 1 + b(floor(k/2)) + b(floor(k/3)) with b(0) = 0, i.e., in A061984.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 2, 1, 3, 0, 0, 4, 2, 0, 0, 6, 0, 0, 0, 3, 5, 4, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 6, 0, 0, 0, 10, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 15, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0

Offset: 0

Henry Bottomley, May 24 2001

nonn

If n is not in A061985 then a(n)=0, otherwise if n=A061985(m) then a(n) = A061987(m).

A088468 a(0) = 1, a(n) = a(floor(n/2)) + a(floor(n/3)) for n > 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 7, 7, 8, 9, 9, 9, 12, 12, 12, 12, 13, 13, 16, 16, 16, 16, 16, 16, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 22, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 33, 33, 33, 33, 33, 33, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 38, 38, 38, 38, 38, 38, 38, 38, 48

Offset: 0

Michael Somos, Oct 02 2003

nonn

Record values greater than 1 occur at 3-smooth numbers: A160519(n)=a(A003586(n)) and A160519(m)A003586(n). - Reinhard Zumkeller, May 16 2009

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
A. R. Lebeck, CPS 104: Homework #3.
Michael Penn, Erdős but simpler, Youtube video.

Equals A061984(n) + 1.

Cf. A007731, A083662, A165704, A165706. - Reinhard Zumkeller, Sep 26 2009

a[0]=1;a[n_]:=a[n]=a[Floor[n/2]]+a[Floor[n/3]];Array[a,75,0] (* Harvey P. Dale, Aug 23 2020 *)

PARI
```
a(n)=if(n<1,n==0,a(n\2)+a(n\3))
```

Limit_{n->oo} a(n)/n = 0, as proved in Michael Penn's Youtube video (see Links). Michael Penn states in the video that this is a simplification of a problem of Paul Erdős, where the original problem is to show that limit_{n->oo} b(n)/n = 12/log(432) for b(0) = 1, b(n) = b(floor(n/2)) + b(floor(n/3)) + b(floor(n/6)) for n > 0 ({b(n)} is the sequence A007731). - Jianing Song, Sep 27 2023

A007731 a(n) = a(floor(n/2)) + a(floor(n/3)) + a(floor(n/6)), with a(0) = 1.

Original entry on oeis.org

1, 3, 5, 7, 9, 9, 15, 15, 17, 19, 19, 19, 29, 29, 29, 29, 31, 31, 41, 41, 41, 41, 41, 41, 55, 55, 55, 57, 57, 57, 57, 57, 59, 59, 59, 59, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 103, 103, 103, 103, 103, 103, 117, 117

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..10000
P. Erdős, A. Hildebrand, A. Odlyzko, P. Pudaite and B. Reznick, The asymptotic behavior of a family of sequences, Pacific J. Math., 126 (1987), pp. 227-241.

Cf. A083662, A088468, A165704, A165706, A061984.

a007731 n = a007731_list !! n
a007731_list = 1 : (zipWith3 (\u v w -> u + v + w)
   (map (a007731 . (`div` 2)) [1..])
   (map (a007731 . (`div` 3)) [1..])
   (map (a007731 . (`div` 6)) [1..]))
-- Reinhard Zumkeller, Jan 11 2014

A007731 := proc(n) option remember; if n=0 then RETURN(1) else RETURN( A007731(trunc(n/2))+A007731(trunc(n/3))+A007731(trunc(n/6))); fi; end;
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1,
      add(a(floor(n/i)), i=[2, 3, 6]))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Sep 27 2023

a[n_] := a[n] = a[Floor[n/2]] + a[Floor[n/3]] + a[Floor[n/6]] ; a[0] = 1; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Mar 06 2014 *)

a(n)=if(n<5, 2*n+1, a(n\2) + a(n\3) + a(n\6)) \\ Charles R Greathouse IV, Feb 08 2017

From given link, a(n) is asymptotic to c*n where c = 12/log(432) = 1.97744865... - Benoit Cloitre, Dec 18 2002

Name clarified by Michel Marcus, Apr 10 2025

A061980 Square array A(n,k) = A(n-1,k) + A(n-1, floor(k/2)) + A(n-1, floor(k/3)), with A(0,0) = 1, read by antidiagonals.

Original entry on oeis.org

1, 0, 3, 0, 2, 9, 0, 1, 8, 27, 0, 0, 6, 26, 81, 0, 0, 4, 23, 80, 243, 0, 0, 3, 20, 76, 242, 729, 0, 0, 3, 17, 72, 237, 728, 2187, 0, 0, 1, 17, 66, 232, 722, 2186, 6561, 0, 0, 1, 11, 66, 222, 716, 2179, 6560, 19683, 0, 0, 1, 11, 54, 222, 701, 2172, 6552, 19682, 59049

Offset: 0

Henry Bottomley, May 24 2001

			Array begins as:
    1,   0,   0,   0,   0,   0,   0, ...;
    3,   2,   1,   0,   0,   0,   0, ...;
    9,   8,   6,   4,   3,   3,   1, ...;
   27,  26,  23,  20,  17,  17,  11, ...;
   81,  80,  76,  72,  66,  66,  54, ...;
  243, 242, 237, 232, 222, 222, 202, ...;
  729, 728, 722, 716, 701, 701, 671, ...;
Antidiagonal rows begin as:
  1;
  0, 3;
  0, 2, 9;
  0, 1, 8, 27;
  0, 0, 6, 26, 81;
  0, 0, 4, 23, 80, 243;
  0, 0, 3, 20, 76, 242, 729;
  0, 0, 3, 17, 72, 237, 728, 2187;
  0, 0, 1, 17, 66, 232, 722, 2186, 6561;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Row sums are 6^n: A000400.
Columns are A000244, A024023, A060188, A061981, A061982 twice, A061983 twice, etc.
Cf. A000244, A061290, A061930, A061979, A061984, A061987.

A[n_, k_]:= A[n, k]= If[n==0, Boole[k==0], A[n-1,k] +A[n-1,Floor[k/2]] +A[n-1, Floor[k/3]]];
T[n_, k_]:= A[k, n-k];
Table[A[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 18 2022 *)

@CachedFunction
def A(n,k):
    if (n==0): return 0^k
    else: return A(n-1, k) + A(n-1, (k//2)) + A(n-1, (k//3))
def T(n, k): return A(k, n-k)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 18 2022

A(n,k) = A(n-1,k) + A(n-1, floor(k/2)) + A(n-1, floor(k/3)), with A(0,0) = 1.
T(n, k) = A(k, n-k).
Sum_{k=0..n} A(n, k) = A000400(n).
T(n, n) = A(n, 0) = A000244(n). - G. C. Greubel, Jun 18 2022

A385911 If n = 2^b3^c, then a(n) = (-1)^b binomial(b+c, b), else a(n) = 0, for n >= 1.

Original entry on oeis.org

1, -1, 1, 1, 0, -2, 0, -1, 1, 0, 0, 3, 0, 0, 0, 1, 0, -3, 0, 0, 0, 0, 0, -4, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 0

Offset: 1

Paul D. Hanna, Jul 12 2025

sign

See comment by David Wasserman in related sequence A061984.
Triangle A385910 has g.f. A(x,y) where A(x,y) = A(x^3 + 3*x*y*A(x,y)^3, y) / A(x^2 + 2*x*y*A(x,y)^2, y).
a(n) = A385910(n+1, 1) for n >= 1.

			G.f. A(x) = x - x^2 + x^3 + x^4 - 2*x^6 - x^8 + x^9 + 3*x^12 + x^16 - 3*x^18 - 4*x^24 + x^27 - x^32 + 6*x^36 + 5*x^48 - 4*x^54 + x^64 - 10*x^72 + x^81 - 6*x^96 + ...
where A(x) equals the sum of the following series
A(x) = (x + x^3 + x^9 + x^27 + ... + x^(3^k) + ...)
  - (x^2 + 2*x^6 + 3*x^18 + 4*x^54 + ... + (k+1)*x^(2*3^k) + ...)
  + (x^4 + 3*x^12 + 6*x^36 + 10*x^108 + ... + C(2+k,k)*x^(2^2*3^k) + ...)
  - (x^8 + 4*x^24 + 10*x^72 + 20*x^216 + ... + C(3+k,k)*x^(2^3*3^k) + ...)
  + (x^16 + 5*x^48 + 15*x^144 + 35*x^432 + ... + C(4+k,k)*x^(2^4*3^k) + ...)
  + ... + (-1)^n * Sum_{k>=0} binomial(n+k,k) * x^(2^n*3^k) + ...

Cf. A385910, A061984.

{a(n) = my(p2,p3); if(n<1,0, p2 = valuation(n,2); p3 = valuation(n,3);
if(n/(2^p2*3^p3)>1,0, (-1)^p2 * binomial(p2 + p3, p2) ))}
for(n=1,100,print1(a(n),", "))

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x) = Sum_{n>=0} (-1)^n * Sum_{k>=0} binomial(n+k,k) * x^(2^n*3^k).
(2) If n = 2^b*3^c, then a(n) = (-1)^b * binomial(b+c, b) else a(n) = 0, for n >= 1.

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A061985 Values which can occur in sequence b(k) = 1 + b(floor(k/2)) + b(floor(k/3)) with b(0) = 0, i.e., in A061984.

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A061986 Number of times n appears in sequence b(k) = 1 + b(floor(k/2)) + b(floor(k/3)) with b(0) = 0, i.e., in A061984.

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A088468 a(0) = 1, a(n) = a(floor(n/2)) + a(floor(n/3)) for n > 0.

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A007731 a(n) = a(floor(n/2)) + a(floor(n/3)) + a(floor(n/6)), with a(0) = 1.

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A061980 Square array A(n,k) = A(n-1,k) + A(n-1, floor(k/2)) + A(n-1, floor(k/3)), with A(0,0) = 1, read by antidiagonals.

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A385911 If n = 2^b*3^c, then a(n) = (-1)^b * binomial(b+c, b), else a(n) = 0, for n >= 1.

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A385911 If n = 2^b3^c, then a(n) = (-1)^b binomial(b+c, b), else a(n) = 0, for n >= 1.