A302792 -id:A302792 - cp's OEIS frontend

A036554 Numbers whose binary representation ends in an odd number of zeros.

2, 6, 8, 10, 14, 18, 22, 24, 26, 30, 32, 34, 38, 40, 42, 46, 50, 54, 56, 58, 62, 66, 70, 72, 74, 78, 82, 86, 88, 90, 94, 96, 98, 102, 104, 106, 110, 114, 118, 120, 122, 126, 128, 130, 134, 136, 138, 142, 146, 150, 152, 154, 158, 160, 162, 166, 168, 170, 174

Offset: 1

Tom Verhoeff

Fraenkel (2010) called these the "dopey" numbers.
Also n such that A035263(n)=0 or A050292(n) = A050292(n-1).
Indices of even numbers in A033485. - Philippe Deléham, Mar 16 2004
a(n) is an odious number (see A000069) for n odd; a(n) is an evil number (see A001969) for n even. - Philippe Deléham, Mar 16 2004
Indices of even numbers in A007913, in A001511. - Philippe Deléham, Mar 27 2004
This sequence consists of the increasing values of n such that A097357(n) is even. - Creighton Dement, Aug 14 2004
Numbers with an odd number of 2's in their prime factorization (e.g., 8 = 2*2*2). - Mark Dow, Sep 04 2007
Equals the set of natural numbers not in A003159 or A141290. - Gary W. Adamson, Jun 22 2008
Represents the set of CCW n-th moves in the standard Tower of Hanoi game; and terms in even rows of a [1, 3, 5, 7, 9, ...] * [1, 2, 4, 8, 16, ...] multiplication table. Refer to the example. - Gary W. Adamson, Mar 20 2010
Refer to the comments in A003159 relating to A000041 and A174065. - Gary W. Adamson, Mar 21 2010
If the upper s-Wythoff sequence of s is s, then s=A036554. (See A184117 for the definition of lower and upper s-Wythoff sequences.) Starting with any nondecreasing sequence s of positive integers, A036554 is the limit when the upper s-Wythoff operation is iterated.  For example, starting with s=(1,4,9,16,...) = (n^2), we obtain lower and upper s-Wythoff sequences
  a=(1,3,4,5,6,8,9,10,11,12,14,...) = A184427;
  b=(2,7,12,21,31,44,58,74,...) = A184428.
  Then putting s=a and repeating the operation gives
b'=(2,6,8,10,13,17,20,...), which has the same first four terms as A036554. - Clark Kimberling, Jan 14 2011
Or numbers having infinitary divisor 2, or the same, having factor 2 in Fermi-Dirac representation as a product of distinct terms of A050376. - Vladimir Shevelev, Mar 18 2013
Thus, numbers not in A300841 or in A302792. Equally, sequence 2*A300841(n) sorted into ascending order. - Antti Karttunen, Apr 23 2018

			From _Gary W. Adamson_, Mar 20 2010: (Start)
Equals terms in even numbered rows in the following multiplication table:
(rows are labeled 1,2,3,... as with the Towers of Hanoi disks)
   1,  3,  5,  7,  9, 11, ...
   2,  6, 10, 14, 18, 22, ...
   4, 12, 20, 28, 36, 44, ...
   8, 24, 40, 56, 72, 88, ...
   ...
As shown, 2, 6, 8, 10, 14, ...; are in even numbered rows, given the row with (1, 3, 5, 7, ...) = row 1.
The term "5" is in an odd row, so the 5th Towers of Hanoi move is CW, moving disc #1 (in the first row).
a(3) = 8 in row 4, indicating that the 8th Tower of Hanoi move is CCW, moving disc #4.
A036554 bisects the positive nonzero natural numbers into those in the A036554 set comprising 1/3 of the total numbers, given sufficiently large n.
This corresponds to 1/3 of the TOH moves being CCW and 2/3 CW. Row 1 of the multiplication table = 1/2 of the natural numbers, row 2 = 1/4, row 3 = 1/8 and so on, or 1 = (1/2 + 1/4 + 1/8 + 1/16 + ...). Taking the odd-indexed terms of this series given offset 1, we obtain 2/3 = 1/2 + 1/8 + 1/32 + ..., while sum of the even-indexed terms is 1/3. (End)

T. D. Noe, Table of n, a(n) for n = 1..1000
L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Representations for a special sequence, Fib. Quart., 10 (1972), 499-518, 550 (see d(n) on page 501).
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
A. S. Fraenkel, Home Page
Aviezri S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.
Aviezri S. Fraenkel, The vile, dopey, evil and odious game players, Discrete Math. 312 (2012), no. 1, 42-46.
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
Eric Sopena, i-Mark: A new subtraction division game, arXiv:1509.04199 [cs.DM], 2015.
M. Stoll, Chabauty without the Mordell-Weil group, arXiv preprint arXiv:1506.04286 [math.NT], 2015.
Index entries for 2-automatic sequences.

Indices of odd numbers in A007814. Subsequence of A036552. Complement of A003159. Also double of A003159.

Cf. A000041, A003157, A003158, A005408, A052330, A072939, A079523, A096268 (characteristic function, when interpreted with offset 1), A141290, A174065, A300841.

a036554 = (+ 1) . a079523  -- Reinhard Zumkeller, Mar 01 2012

[2*m:m in [1..100] | Valuation(m,2) mod 2 eq 0]; // Marius A. Burtea, Aug 29 2019

Select[Range[200],OddQ[IntegerExponent[#,2]]&] (* Harvey P. Dale, Oct 19 2011 *)

is(n)=valuation(n,2)%2 \\ Charles R Greathouse IV, Nov 20 2012

def ok(n):
  c = 0
  while n%2 == 0: n //= 2; c += 1
  return c%2 == 1
print([m for m in range(1, 175) if ok(m)]) # Michael S. Branicky, Feb 06 2021

from itertools import count, islice
def A036554_gen(startvalue=1): return filter(lambda n:(~n & n-1).bit_length()&1,count(max(startvalue,1))) # generator of terms >= startvalue
A036554_list = list(islice(A036554_gen(),30)) # Chai Wah Wu, Jul 05 2022

is_A036554 = lambda n: A001511(n)&1==0 # M. F. Hasler, Nov 26 2024

def A036554(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, s = n+x, bin(x)[2:]
        l = len(s)
        for i in range(l&1,l,2):
            c -= int(s[i])+int('0'+s[:i],2)
        return c
    return bisection(f,n,n) # Chai Wah Wu, Jan 29 2025

a(n) = A079523(n)+1 = A072939(n)-1.
a(n) = A003156(n) + n = A003157(n) - n = A003158(n) - n + 1. - Philippe Deléham, Apr 10 2004
Values of k such that A091297(k) = 2. - Philippe Deléham, Feb 25 2004
a(n) ~ 3n. - Charles R Greathouse IV, Nov 20 2012 [In fact, a(n) = 3n + O(log n). - Charles R Greathouse IV, Nov 27 2024]
a(n) = 2*A003159(n). - Clark Kimberling, Sep 30 2014
{a(n)} = A052330({A005408(n)}), where {a(n)} denotes the set of integers in the sequence. - Peter Munn, Aug 26 2019

Incorrect equation removed from formula by Peter Munn, Dec 04 2020

A223490 Smallest Fermi-Dirac factor of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 7, 2, 9, 2, 11, 3, 13, 2, 3, 16, 17, 2, 19, 4, 3, 2, 23, 2, 25, 2, 3, 4, 29, 2, 31, 2, 3, 2, 5, 4, 37, 2, 3, 2, 41, 2, 43, 4, 5, 2, 47, 3, 49, 2, 3, 4, 53, 2, 5, 2, 3, 2, 59, 3, 61, 2, 7, 4, 5, 2, 67, 4, 3, 2, 71, 2, 73, 2, 3, 4, 7, 2, 79

Offset: 1

Reinhard Zumkeller, Mar 20 2013

Note that this is not equal to the smallest Fermi-Dirac prime (A050376) dividing n, which is always A020639(n). - Antti Karttunen, Apr 15 2018

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
OEIS Wiki, "Fermi-Dirac representation" of n

Cf. A223491, A050376, A028233, A000040 (subsequence).
Cf. A302023, A302024, A302786, A302792.
Cf. also A020639.

Haskell
```
a223490 = head . a213925_row
```

f[p_, e_] := p^(2^IntegerExponent[e, 2]); a[n_] := Min @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 26 2020 *)

up_to = 65537;
v050376 = vector(up_to);
A050376(n) = v050376[n];
ispow2(n) = (n && !bitand(n,n-1));
i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to,break));
A052331(n) = { my(s=0,e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); };
A001511(n) = 1+valuation(n,2);
A223490(n) = if(1==n,n,A050376(A001511(A052331(n)))); \\ Antti Karttunen, Apr 15 2018

a(n) = A213925(n,1).
A209229(A100995(a(n))) = 1; A010055(a(n)) = 1.
From Antti Karttunen, Apr 15 2018: (Start)
a(1) = 1; and for n > 1, a(n) = A050376(A302786(n)).
a(n) = n / A302792(n).
a(n) = A302023(A020639(A302024(n))).
(End)

A302024 Permutation of natural numbers mapping "Fermi-Dirac factorization" to ordinary factorization: a(1) = 1, a(2A300841(n)) = 2a(n), a(A300841(n)) = A003961(a(n)).

Original entry on oeis.org

1, 2, 3, 5, 7, 4, 11, 6, 13, 10, 17, 9, 19, 14, 15, 23, 29, 22, 31, 25, 21, 26, 37, 8, 41, 34, 33, 35, 43, 12, 47, 38, 39, 46, 49, 55, 53, 58, 51, 18, 59, 20, 61, 65, 77, 62, 67, 57, 71, 74, 69, 85, 73, 28, 91, 30, 87, 82, 79, 27, 83, 86, 121, 95, 119, 44, 89, 115, 93, 50, 97, 42, 101, 94, 111, 145, 143, 52, 103, 133, 107, 106, 109, 45, 161

Offset: 1

Antti Karttunen, Apr 15 2018

nonn

Because "Fermi-Dirac factorization" is fundamentally different from ordinary prime factorization (as no exponents larger than 1 are allowed) this pair of permutations mapping between them is not always very intuitive. For example, we have ("as expected") A302776(n) = A302023(A052126(A302024(n))), while on the other hand, we have A302792(n) = A300841(A302023(A032742(A302024(n)))), where an additional shift-operator A300841 is needed for "correction".

Cf. A302023 (inverse).

Cf. A050376, A052331, A003961, A005940, A300840, A300841, A302791.

up_to = 32768;
v050376 = vector(up_to);
A050376(n) = v050376[n];
ispow2(n) = (n && !bitand(n,n-1));
i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to,break));
A052331(n) = { my(s=0,e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
A302024(n) = A005940(1+A052331(n));

a(n) = A005940(1+A052331(n)).
a(A050376(n)) = A000040(n).
A001221(a(n)) = A302790(n).
A001222(a(n)) = A064547(n).

A302776 a(1) = 1; for n>1, a(n) = n/(largest Fermi-Dirac factor of n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 3, 1, 1, 2, 1, 4, 3, 2, 1, 6, 1, 2, 3, 4, 1, 6, 1, 2, 3, 2, 5, 4, 1, 2, 3, 8, 1, 6, 1, 4, 5, 2, 1, 3, 1, 2, 3, 4, 1, 6, 5, 8, 3, 2, 1, 12, 1, 2, 7, 4, 5, 6, 1, 4, 3, 10, 1, 8, 1, 2, 3, 4, 7, 6, 1, 5, 1, 2, 1, 12, 5, 2, 3, 8, 1, 10, 7, 4, 3, 2, 5, 6, 1, 2, 9, 4, 1, 6, 1, 8, 15

Offset: 1

Antti Karttunen, Apr 13 2018

nonn

For n > 1, a(n) = the smallest positive number d such that n/d is a "Fermi-Dirac prime", a term of A050376.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A050376, A052331, A223491, A302023, A302024, A302777, A302792.
Cf. A084400 (gives the positions of 1's).
Cf. also A052126, A284600.

f[p_, e_] := p^(2^Floor[Log2[e]]); a[n_] := n / Max @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 26 2020 *)

A209229(n) = (n && !bitand(n,n-1));
A302777(n) = A209229(isprimepower(n));
A302776(n) = if(1==n,n,fordiv(n, d, if(A302777(n/d), return(d))));

a(n) = n / A223491(n).

a(n) = A302023(A052126(A302024(n))).

A365296 The smallest coreful infinitary divisor of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 2, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 6, 25, 26, 3, 28, 29, 30, 31, 2, 33, 34, 35, 36, 37, 38, 39, 10, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 6, 55, 14, 57, 58, 59, 60, 61, 62, 63, 4, 65, 66, 67, 68, 69

Offset: 1

Amiram Eldar, Aug 31 2023

A coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n.
The number of coreful infinitary divisors of n is A363329(n).
All the terms are in A138302.

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

Cf. A006519, A037445, A077609, A007947, A034444, A138302, A268335, A302792, A339597, A363329.

f[p_, e_] := p^(2^IntegerExponent[e, 2]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(2^valuation(f[i,2], 2)));}

from math import prod
from sympy import factorint
def A365296(n): return prod(p**(e&-e) for p, e in factorint(n).items()) # Chai Wah Wu, Sep 01 2023

Multiplicative with a(p^e) = p^A006519(e).
a(n) = n if and only if n is in A138302.
a(n) >= A007947(n) with equality if and only if n is an exponentially odd number (A268335).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - 1/p + Sum_{e>=1} 1/p^f(e)-1/p^(f(e)+1)) = 0.4459084041..., where f(k) = 2*k - A006519(k) = A339597(k-1).
A037445(a(n)) = A034444(n). - Amiram Eldar, Oct 19 2023

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A036554 Numbers whose binary representation ends in an odd number of zeros.

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A223490 Smallest Fermi-Dirac factor of n.

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A302024 Permutation of natural numbers mapping "Fermi-Dirac factorization" to ordinary factorization: a(1) = 1, a(2*A300841(n)) = 2*a(n), a(A300841(n)) = A003961(a(n)).

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A302776 a(1) = 1; for n>1, a(n) = n/(largest Fermi-Dirac factor of n).

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A365296 The smallest coreful infinitary divisor of n.

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A302024 Permutation of natural numbers mapping "Fermi-Dirac factorization" to ordinary factorization: a(1) = 1, a(2A300841(n)) = 2a(n), a(A300841(n)) = A003961(a(n)).