A050292 -id:A050292 - cp's OEIS frontend

A177334 Largest factor in the factorization of n! over distinct terms of A050376.

2, 3, 4, 5, 16, 16, 16, 81, 256, 256, 256, 256, 256, 256, 256, 256, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 4294967296, 4294967296, 4294967296, 4294967296, 4294967296, 4294967296

Offset: 2

Vladimir Shevelev, May 06 2010

nonn

Each number >=2 has a unique factorization over distinct terms of A050376.
This is obtained from the standard prime factor representation by splitting the exponents into a sum of powers of 2, and further factorization according to the nonzero term of this base-2 representation.
The largest factor of this representation of A000142(n) defines this sequence.

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 [Russian].

Amiram Eldar, Table of n, a(n) for n = 2..1000
S. Litsyn and V. S. Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.

Cf. A050376, A177329, A055460, A177333, A176525, A001358, A176509, A064380, A050292.

A177334 := proc(n) local a,p,pow2 ; a := 1 ; for p in ifactors(n!)[2] do pow2 := convert( op(2,p),base,2) ; for j from 1 to nops(pow2) do if op(j,pow2) <> 0 then a := max(a,op(1,p)^(2^(j-1))) ; end if; end do: end do: return a ; end proc:
seq(A177334(n),n=2..60) ; # R. J. Mathar, Jun 16 2010

b[n_] :=2^(-1+Position[ Reverse@IntegerDigits[n, 2],?(#==1&)])//Flatten; a[n] := Module[{np = PrimePi[n]}, v=Table[0,{np}]; Do[p = Prime[k]; Do[v[[k]] += IntegerExponent[j, p], {j,2,n}],  {k,1,np}]; Max[(Prime/@Range[np])^(b/@v) // Flatten]]; Array[a, 38, 2]  (* Amiram Eldar, Sep 17 2019 *)

a(18) and a(19) corrected and sequence extended by R. J. Mathar, Jun 16 2010

A120385 If a(n-1) = 1 then largest value so far + 1, otherwise floor(a(n-1)/2); or table T(n,k) with T(n,0) = n, T(n,k+1) = floor(T(n,k)/2).

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 2, 1, 5, 2, 1, 6, 3, 1, 7, 3, 1, 8, 4, 2, 1, 9, 4, 2, 1, 10, 5, 2, 1, 11, 5, 2, 1, 12, 6, 3, 1, 13, 6, 3, 1, 14, 7, 3, 1, 15, 7, 3, 1, 16, 8, 4, 2, 1, 17, 8, 4, 2, 1, 18, 9, 4, 2, 1, 19, 9, 4, 2, 1, 20, 10, 5, 2, 1, 21, 10, 5, 2, 1, 22, 11, 5, 2, 1, 23, 11, 5, 2, 1, 24, 12, 6, 3, 1, 25

Offset: 1

Franklin T. Adams-Watters, Jun 29 2006

Although not strictly a fractal sequence as defined in the Kimberling link, this sequence has many fractal properties. If the first instance of each value is removed, the result is the original sequence with each row repeated twice. Removing all odd-indexed instances of each value does give the original sequence.

			The table starts:
  1;
  2, 1;
  3, 1;
  4, 2, 1;
  5, 2, 1;
  6, 3, 1;
  7, 3, 1;
  8, 4, 2, 1;

Alois P. Heinz, Rows n = 1..2048, flattened
C. Kimberling, Fractal sequences

Cf. A029837 (row lengths), A083652 (position of first n).
Cf. A005187 (row sums). A001477, A050292, A080277.
Cf. A000027, A004526, A002265, A132292.

T:= proc(n) T(n):= `if`(n=1, 1, [n, T(iquo(n, 2))][]) end:
seq(T(n), n=1..30);  # Alois P. Heinz, Feb 12 2019

Flatten[Function[n,NestWhile[Append[#, Floor[Last[#]/2]] &, {n}, Last[#] != 1 &]][#] & /@ Range[50]] (* Birkas Gyorgy, Apr 14 2011 *)

T(n,k) = floor(n/2^(k-1)).
From Peter Bala, Feb 02 2013: (Start)
The n-th row polynomial R(n,t) = Sum_{k>=0} t^k*floor(n/2^k) and satisfies the recurrence equation R(n,t) = t*R(floor(n/2),t) + n, with R(1,t) = 1.
O.g.f. Sum_{n>=1} R(n,t)*x^n = 1/(1-x)*Sum_{n>=0} t^n*x^(2^n)/(1 - x^(2^n)).
Product_{n>=1} ( 1 + x^((t^n - 2^n)/(t-2)) ) = 1 + Sum_{n>=1} x^R(n,t) = 1 + x + x^(2 + t) + x^(3 + t) + x^(4 + 2*t + t^2) + .... For related sequences see A050292 (t = -1), A001477(t = 0), A005187 (t = 1) and A080277 (t = 2).
(End)

A176525 Fermi-Dirac semiprimes: products of two distinct terms of A050376.

Original entry on oeis.org

6, 8, 10, 12, 14, 15, 18, 20, 21, 22, 26, 27, 28, 32, 33, 34, 35, 36, 38, 39, 44, 45, 46, 48, 50, 51, 52, 55, 57, 58, 62, 63, 64, 65, 68, 69, 74, 75, 76, 77, 80, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 100, 106, 111, 112, 115, 116, 117, 118, 119, 122

Offset: 1

Vladimir Shevelev, Apr 19 2010, Apr 20 2010

nonn

The sequence essentially differs from A000379 beginning with a(108)=212 (not 210). All squarefree terms of A001358 are in the sequence.

Vladimir S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences, Vol. 4 (1996), pp. 28-43 [Russian].

Peter J. C. Moses, Table of n, a(n) for n = 1..10000
Simon Litsyn and Vladimir S. Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, Vol. 7 (2007), Article #A33, pp. 1-36.

Cf. A001358, A050376, A000379, A176472, A176509, A064380, A050292.

Select[Range[120], Plus @@ DigitCount[Last /@ FactorInteger[#], 2, 1] == 2 &] (* Amiram Eldar, Nov 27 2020 *)

If a(n)=u*v, uA050376 "Fermi-Dirac primes", then A064380(a(n))=a(n)-u-v+1+Sum{i>=1}(-1)^(i-1)*floor(v/u^i).

Effectively duplicate content (due to duplicate referenced sequence) removed by Peter Munn, Dec 19 2019

A197911 Representable by A001045 (Jacobsthal sequence). Complement of A003158.

Original entry on oeis.org

0, 1, 3, 4, 5, 6, 8, 9, 11, 12, 14, 15, 16, 17, 19, 20, 21, 22, 24, 25, 26, 27, 29, 30, 32, 33, 35, 36, 37, 38, 40, 41, 43, 44, 46, 47, 48, 49, 51, 52, 54, 55, 57, 58, 59, 60, 62, 63, 64, 65, 67, 68, 69, 70, 72, 73, 75, 76, 78, 79, 80, 81, 83, 84, 85, 86, 88

Offset: 0

Philippe Deléham, Oct 19 2011

a(n) modulo 2 is the Prouhet-Thue-Morse sequence A010060.
The sequence of Jacobsthal numbers A001045 begins [1, 1, 3, 5, 11, 21, ...] with two occurrences of the term 1. Allowing for this, we find that the numbers representable as a sum of distinct Jacobsthal numbers form A050292. - Peter Bala, Feb 02 2013
Partial sums of A056832. - Reinhard Zumkeller, Jul 29 2014

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Representations for a special sequence, Fib. Quart., 10 (1972), 499-518, 550 (see page 499).

Cf. A001045, A003158, A010060. A050292.

Cf. A056832.

a197911 n = a197911_list !! n
a197911_list = scanl (+) 0 a056832_list
-- Reinhard Zumkeller, Jul 29 2014

def A197911(n): return n+sum((~(i+1)&i).bit_length()&1 for i in range(n)) # Chai Wah Wu, Jan 09 2023

a(n) = Sum_{k>=0} A030308(n,k)*A001045(k+2).

A068639 a(0) = 0, a(n) = a(n-1) + (-1)^p(n) for n >= 1, where p(n) = highest power of 2 dividing n.

Original entry on oeis.org

0, 1, 0, 1, 2, 3, 2, 3, 2, 3, 2, 3, 4, 5, 4, 5, 6, 7, 6, 7, 8, 9, 8, 9, 8, 9, 8, 9, 10, 11, 10, 11, 10, 11, 10, 11, 12, 13, 12, 13, 12, 13, 12, 13, 14, 15, 14, 15, 16, 17, 16, 17, 18, 19, 18, 19, 18, 19, 18, 19, 20, 21, 20, 21, 22, 23, 22, 23, 24, 25, 24, 25, 24, 25, 24, 25, 26, 27, 26

Offset: 0

N. J. A. Sloane, Oct 01 2003

Paolo Xausa, Table of n, a(n) for n = 0..10000
J.-P. Allouche and Jeffrey Shallit, The Ring of k-regular Sequences, II
J.-P. Allouche and Jeffrey Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions

Cf. A050292, A065359, A087733.

A068639[n_] := (n + 2*Total[(-1)^Range[0, Max[0, BitLength[n] - 1]]*Reverse[IntegerDigits[n, 2]]])/3;
Array[A068639, 101, 0] (* or *)
Join[{0}, Accumulate[(-1)^IntegerExponent[Range[100], 2]]] (* Paolo Xausa, Jun 05 2025 *)

PARI
```
a(n)=if(n<1,0,ceil(n/2)-a(n-ceil(n/2)))
```

a(n) = (n+2*A065359(n))/3; a(n) is asymptotic to n/3. - Benoit Cloitre, Oct 04 2003
From Ralf Stephan, Oct 17 2003: (Start)
a(0)=0, a(2n) = -a(n) + n, a(2n+1) = -a(n) + n + 1.
a(n) = (1/2) * (A050292(n) + A065359(n)).
G.f.: (1/2) * 1/(1-x) * Sum_{k>=0} (-1)^k*t/(1-t^2) where t=x^2^k. (End)
a(0)=0 then a(n) = ceiling(n/2)-a(n-ceiling(n/2)). - Benoit Cloitre, May 03 2004

More terms from John W. Layman and Robert G. Wilson v, Oct 02 2003

A123087 Sequence of numbers such that a(2*n) + a(n) = n and a(n) is the smallest number such that a(n) >= a(n-1).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 11, 11, 12, 12, 12, 12, 13, 13, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 24, 24, 25, 25, 25, 25, 26

Offset: 0

Philippe Deléham, Sep 27 2006

nonn

If the value a(n) = m >= 1 is appearing for the first time, then n is of the form n = 2^k*s, where k,s are odd numbers. Therefore every m occurs 2 or 4 times consecutively. More exactly, if n+2 has the same form as n (i.e., 2^k*s with odd k,s), then a(n) = m occurs 2 times, otherwise, m occurs 4 times. - Vladimir Shevelev, Aug 25 2010

a(n) is the number of those numbers not exceeding n for which 2 is an infinitary divisor (for definition see comment at A037445). - Vladimir Shevelev, Feb 21 2011

			  a(2*0) + a(0) = 0 -----> a(0)  = 0
  a(1)  >= a(0) ---------> a(1)  = 0
  a(2*1) + a(1) = 1 -----> a(2)  = 1
  a(3)  >= a(2) ---------> a(3)  = 1
  a(2*2) + a(2) = 2 -----> a(4)  = 1
  a(5)  >= a(4) ---------> a(5)  = 1
  a(2*3) + a(3) = 3 -----> a(6)  = 2
  a(7)  >= a(6) ---------> a(7)  = 2
  a(2*4) + a(4) = 4 -----> a(8)  = 3
  a(9)  >= a(8) ---------> a(9)  = 3
  a(2*5) + a(5) = 5 -----> a(10) = 4
  a(11) >= a(10) --------> a(11) = 4
  a(2*6) + a(6) = 6 -----> a(12) = 4
  a(13) >= a(12) --------> a(13) = 4
  a(2*7) + a(7) = 7 -----> a(14) = 5

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Partial sums of A096268 and of A328979.

Cf. A050292, A065359.

a123087 n = a123087_list !! n
a123087_list = scanl (+) 0 a096268_list
-- Reinhard Zumkeller, Jul 29 2014

a(n)=if(n<1,0,floor(n/2)-a(floor(n/2))) \\ Benoit Cloitre, Sep 30 2006

a(0)=0, a(n) = floor(n/2) - a(floor(n/2)); partial sums of A096268; a(2n) = A050292(n); a(n) is asymptotic to n/3. - Benoit Cloitre, Sep 30 2006
a(2*n+1) = a(2*n); a(n) = n/3 + O(log(n)), moreover, the equation a(3m) = m has infinitely many solutions, e.g., a(3*2^k) = 2^k; on the other hand, a((4^k-1)/3) = (4^k-1)/9 - k/3, i.e., limsup|a(n) - n/3| = infinity. - Vladimir Shevelev, Aug 25 2010
a(n) = (n - A065359(n))/3. - Velin Yanev, Jul 13 2021
a(n) = n - A050292(n). - Max Alekseyev, Mar 05 2023

A076895 a(1) = 1, a(n) = n - a(ceiling(n/2)).

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 4, 5, 6, 7, 7, 8, 9, 10, 10, 11, 11, 12, 12, 13, 14, 15, 15, 16, 16, 17, 17, 18, 19, 20, 20, 21, 22, 23, 23, 24, 25, 26, 26, 27, 27, 28, 28, 29, 30, 31, 31, 32, 33, 34, 34, 35, 36, 37, 37, 38, 38, 39, 39, 40, 41, 42, 42, 43, 43, 44, 44, 45, 46, 47, 47, 48, 48

Offset: 1

Benoit Cloitre, Nov 26 2002

nonn

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

Cf. A050292, A079411.

a[1] = 1; a[n_] := a[n] = n - a[Ceiling[n/2]]; Array[a, 100] (* Amiram Eldar, Sep 10 2020 *)

PARI
```
a(n)=if(n<2,1,n-a(ceil(n/2)))
```

a(n) is asymptotic to (2/3)*n.
a(n) = A050292(n-1) + (1+(1-2*(C(n)-F(n)))*(-1)^F(n))/2 where C(n) = ceiling(log_2(n)); F(n) = floor(log_2(n)) and A050292(n) (with A050292(0)=0) is the maximum cardinality of a double-free subset of {1, 2, ..., n}. So using Wang's asymptotic formula for A050292, a(n) = (2/3)*n + O(log_4(n)). Series expansion: (1/(x-1)) * ( 1/(x-1) + Sum_{i>=1} (-1)^i*( x^(2^i)/(x^(2^i)-1) - x^(2^(i-1)) ) ). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 17 2003
a(n+1) = b(n) with b(0)=0, b(2n) = -b(n) + 2n+1, b(2n+1) = -b(n) + 2n + 2 -[n==0]. Also a(n+1) = A050292(n) + A030301(n). - Ralf Stephan, Oct 28 2003

Edited by Ralf Stephan, Sep 01 2004

A357378 Lexicographically earliest sequence of positive integers such that the values a(floor(n/2)) * a(n) are all distinct.

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 1, 3, 3, 4, 1, 3, 7, 11, 6, 7, 8, 9, 5, 7, 13, 14, 10, 11, 5, 6, 2, 4, 6, 8, 7, 8, 4, 5, 5, 6, 5, 10, 9, 10, 2, 3, 6, 7, 6, 8, 5, 6, 13, 15, 12, 13, 17, 19, 13, 16, 15, 16, 11, 13, 11, 13, 14, 15, 17, 19, 17, 19, 20, 21, 17, 18, 22, 23, 13

Offset: 0

Rémy Sigrist, Sep 26 2022

nonn

See A357379 for the corresponding products.
The sequence is well defined; we can always extend it with a value strictly greater than the square of the greatest value so far.
The sequence is unbounded (otherwise we could only have a finite number of products a(floor(n/2)) * a(n), and therefore a finite number of terms).
For any prime number p: the first term >= p is p.
All prime numbers appear in the sequence.
If a(n) is the first occurrence of some prime number p, then a(m) = 1 for some m in the interval floor(n/2)..2*n.
There are infinitely many 1's in the sequence; as a consequence, every positive integer appears in A357379.

			The first terms are:
  n   a(n)  a(floor(n/2))  a(floor(n/2))*a(n)
  --  ----  -------------  ------------------
   0     1              1                   1
   1     2              1                   2
   2     2              2                   4
   3     3              2                   6
   4     4              2                   8
   5     5              2                  10
   6     1              3                   3
   7     3              3                   9
   8     3              4                  12
   9     4              4                  16
  10     1              5                   5
  11     3              5                  15
  12     7              1                   7

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Density plot of the first 10000000 terms
Rémy Sigrist, C program

Cf. A050292 (additive variant), A357379.

C
```
See Links section.
```

from itertools import count, islice
def agen(): # generator of terms
    alst, disallowed = [1], {1}; yield 1
    for n in count(1):
        ahalf, k = alst[n//2], 1
        while ahalf*k in disallowed: k += 1
        an = k; yield an; alst.append(an); disallowed.add(ahalf*an)
print(list(islice(agen(), 75))) # Michael S. Branicky, Sep 26 2022

a(2*n + 1) > a(2*n).

A076896 a(1) = 1, a(n) = n-a(floor(2n/3)).

Original entry on oeis.org

1, 1, 2, 3, 3, 3, 4, 5, 6, 7, 7, 7, 8, 8, 8, 9, 10, 11, 12, 12, 13, 14, 15, 15, 16, 16, 16, 17, 17, 18, 19, 19, 19, 20, 20, 21, 22, 22, 23, 24, 25, 25, 26, 27, 27, 28, 28, 29, 30, 31, 31, 32, 33, 33, 34, 34, 35, 36, 36, 36, 37, 37, 38, 39, 39, 39, 40, 41, 41, 42, 43, 43, 44, 44

Offset: 1

Benoit Cloitre, Nov 26 2002

nonn

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

Cf. A050292.

a[1] = 1; a[n_] := a[n] = n - a[Floor[2*n/3]]; Array[a, 100] (* Amiram Eldar, May 09 2022 *)

a(n)=if(n==0,0,n-a(floor(2*n/3))); \\ Joerg Arndt, Apr 27 2013

a(n) is asymptotic to (6/10)*n.

A335115 a(2n) = 2n - a(n), a(2n+1) = 2n + 1.

Original entry on oeis.org

1, 1, 3, 3, 5, 3, 7, 5, 9, 5, 11, 9, 13, 7, 15, 11, 17, 9, 19, 15, 21, 11, 23, 15, 25, 13, 27, 21, 29, 15, 31, 21, 33, 17, 35, 27, 37, 19, 39, 25, 41, 21, 43, 33, 45, 23, 47, 33, 49, 25, 51, 39, 53, 27, 55, 35, 57, 29, 59, 45, 61, 31, 63, 43, 65, 33, 67, 51, 69, 35, 71, 45, 73, 37, 75

Offset: 1

Ilya Gutkovskiy, May 23 2020

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

Cf. A035263, A050292, A129527, A154269, A001045.

a[n_] := a[n] = If[EvenQ[n], n - a[n/2], n]; Table[a[n], {n, 1, 75}]
nmax = 75; CoefficientList[Series[Sum[(-1)^k x^(2^k)/(1 - x^(2^k))^2, {k, 0, Floor[Log[2, nmax]]}], {x, 0, nmax}], x] // Rest
f[p_, e_] := If[p == 2, (2^(e + 1) + (-1)^e)/3, p^e]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)

a(n) = my(k=valuation(n,2)); (n<<1 + (n>>k)*(-1)^k)/3; \\ Kevin Ryde, Oct 06 2020

G.f.: Sum_{k>=0} (-1)^k * x^(2^k) / (1 - x^(2^k))^2.
G.f. A(x) satisfies: A(x) = x / (1 - x)^2 - A(x^2).
Dirichlet g.f.: zeta(s-1) / (1 + 2^(-s)).
a(n) = Sum_{d|n} A154269(n/d) * d.
Sum_{k=1..n} a(k) ~ 2*n^2/5. - Vaclav Kotesovec, Jun 11 2020
Multiplicative with a(2^e) = A001045(e+1) and a(p^e) = p^e for e >= 0 and prime p > 2. - Werner Schulte, Oct 05 2020

cp's OEIS Frontend

A177334 Largest factor in the factorization of n! over distinct terms of A050376.

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A120385 If a(n-1) = 1 then largest value so far + 1, otherwise floor(a(n-1)/2); or table T(n,k) with T(n,0) = n, T(n,k+1) = floor(T(n,k)/2).

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A176525 Fermi-Dirac semiprimes: products of two distinct terms of A050376.

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A197911 Representable by A001045 (Jacobsthal sequence). Complement of A003158.

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A068639 a(0) = 0, a(n) = a(n-1) + (-1)^p(n) for n >= 1, where p(n) = highest power of 2 dividing n.

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A123087 Sequence of numbers such that a(2*n) + a(n) = n and a(n) is the smallest number such that a(n) >= a(n-1).

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Haskell

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A076895 a(1) = 1, a(n) = n - a(ceiling(n/2)).

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Mathematica

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A335115 a(2n) = 2n - a(n), a(2n+1) = 2n + 1.