A059895 Table a(i,j) = product prime[k]^(Ei[k] AND Ej[k]) where Ei and Ej are the vectors of exponents in the prime factorizations of i and j; AND is the bitwise operation on binary representation of the exponents.

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 2, 1, 1, 5, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 6, 1, 4, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 7, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 5, 1, 3, 1, 1, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1

Offset

1,5

Comments

Analogous to GCD, with AND replacing MIN.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of the array

Formula

From Antti Karttunen, Apr 11 2017: (Start)

A(x,y) = A059896(x,y) / A059897(x,y).

A(x,y) * A059896(x,y) = A(x,y)^2 * A059897(x,y) = x*y.

(End)

Example

The top left 18 X 18 corner of the array:
1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1
1,  2,  1,  1,  1,  2,  1,  2,  1,  2,  1,  1,  1,  2,  1,  1,  1,  2
1,  1,  3,  1,  1,  3,  1,  1,  1,  1,  1,  3,  1,  1,  3,  1,  1,  1
1,  1,  1,  4,  1,  1,  1,  4,  1,  1,  1,  4,  1,  1,  1,  1,  1,  1
1,  1,  1,  1,  5,  1,  1,  1,  1,  5,  1,  1,  1,  1,  5,  1,  1,  1
1,  2,  3,  1,  1,  6,  1,  2,  1,  2,  1,  3,  1,  2,  3,  1,  1,  2
1,  1,  1,  1,  1,  1,  7,  1,  1,  1,  1,  1,  1,  7,  1,  1,  1,  1
1,  2,  1,  4,  1,  2,  1,  8,  1,  2,  1,  4,  1,  2,  1,  1,  1,  2
1,  1,  1,  1,  1,  1,  1,  1,  9,  1,  1,  1,  1,  1,  1,  1,  1,  9
1,  2,  1,  1,  5,  2,  1,  2,  1, 10,  1,  1,  1,  2,  5,  1,  1,  2
1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 11,  1,  1,  1,  1,  1,  1,  1
1,  1,  3,  4,  1,  3,  1,  4,  1,  1,  1, 12,  1,  1,  3,  1,  1,  1
1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 13,  1,  1,  1,  1,  1
1,  2,  1,  1,  1,  2,  7,  2,  1,  2,  1,  1,  1, 14,  1,  1,  1,  2
1,  1,  3,  1,  5,  3,  1,  1,  1,  5,  1,  3,  1,  1, 15,  1,  1,  1
1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 16,  1,  1
1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 17,  1
1,  2,  1,  1,  1,  2,  1,  2,  9,  2,  1,  1,  1,  2,  1,  1,  1, 18
A(864,1944) = A(2^5*3^3,2^3*3^5) = 2^(5 AND 3)* 3^(3 AND 5) = 2^1*3^1 = 6.

Mathematica

a[i_, i_] := i;
a[i_, j_] := Module[{f1 = FactorInteger[i], f2 = FactorInteger[j], e1, e2}, Scan[(e1[#[[1]]] = #[[2]])&, f1]; Scan[(e2[#[[1]]] = #[[2]])&, f2]; Times @@ (#^BitAnd[e1[#], e2[#]]& /@ Intersection[f1[[All, 1]], f2[[All, 1]]]) ];
Table[a[i - j + 1, j], {i, 1, 15}, {j, 1, i}] // Flatten (* Jean-François Alcover, Jun 19 2018 *)

Prog

(Scheme)
(define (A059895 n) (A059895bi (A002260 n) (A004736 n)))
(define (A059895bi a b) (let loop ((a a) (b b) (m 1)) (cond ((= 1 a) m) ((= 1 b) m) ((equal? (A020639 a) (A020639 b)) (loop (A028234 a) (A028234 b) (* m (expt (A020639 a) (A004198bi (A067029 a) (A067029 b)))))) ((< (A020639 a) (A020639 b)) (loop (A028234 a) b m)) (else (loop a (A028234 b) m)))))
;; Antti Karttunen, Apr 11 2017

Crossrefs

Cf. A003985 (A004198), A003989, A028234, A059896, A059897, A067029, A267115, A284578.

Keyword

base,easy,nonn,tabl

Author

Marc LeBrun, Feb 06 2001

Extensions

Data section extended to 120 terms by Antti Karttunen, Apr 11 2017

Status

approved

A014491 a(n) = gcd(n, 2^n - 1).

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 9, 1, 5, 7, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 9, 1, 1, 1, 5, 1, 21, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 27, 1, 1, 1, 1, 1, 15, 1, 1, 7, 1, 1, 3, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 21, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 3, 1, 1

Offset

1,6

Comments

Also the GCD of the "binary n-th powers", the set of positive integers whose base-2 representation consists of a block of bits repeated n times consecutively. - Jeffrey Shallit, Jan 16 2018

prime(k) for k >= 2 divides a(n) if and only if n is divisible by prime(k)*A014664(k). - Robert Israel, Jan 16 2018

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Daniel M. Kane, Carlo Sanna, and Jeffrey Shallit, Waring's theorem for binary powers, arXiv:1801.04483 [math.NT], Jan 13 2018.

Maple

A014491:=n->igcd(n, 2^n-1); seq(A014491(n), n=1..100); # Wesley Ivan Hurt, Feb 02 2014

Mathematica

Table[GCD[n, 2^n-1], {n, 100}] (* Harvey P. Dale, Mar 14 2013 *)

Crossrefs

Cf. A014664.

Keyword

nonn,easy,nice

Author

Gary M. Mcguire (gmm8n(AT)weyl.math.virginia.edu)

Status

approved

A291448 Triangle read by rows, denominators of coefficients (in rising powers) of rational polynomials P(n,x) such that Integral_{x=0..1} P'(n,x) = BernoulliMedian(n).

1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 5, 1, 1, 1, 3, 1, 5, 1, 7, 1, 1, 1, 3, 1, 5, 1, 7, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 11, 1, 1, 1, 3, 1, 5, 1, 7, 1, 1, 1, 11, 1, 13, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 1, 11, 1, 13, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 1, 11, 1, 13, 1, 1

Offset

0,6

Comments

See A291447 and A290694 for comments.

Links

Table of n, a(n) for n=0..87.

Formula

T(n,k) = Denominator([x^k] Integral(Sum_{j=0..n}(-1)^(n-j)*Stirling2(n,j)*j!*x^j)^m) for m = 2, n >= 0 and k = 0..m*n+1.

Example

Triangle starts:
[1, 1]
[1, 1, 1, 3]
[1, 1, 1, 3, 1, 5]
[1, 1, 1, 3, 1, 5, 1, 7]
[1, 1, 1, 3, 1, 5, 1, 7, 1, 1]
[1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 11]
[1, 1, 1, 3, 1, 5, 1, 7, 1, 1, 1, 11, 1, 13]

Maple

# See A291447.

Mathematica

T[n_] := Integrate[Sum[(-1)^(n-j+1) StirlingS2[n, j] j! x^j, {j, 0, n}]^2, x];
Trow[n_] := CoefficientList[T[n], x] // Denominator;
Table[Trow[r], {r, 0, 7}] // Flatten

Crossrefs

Cf. A164555/A027642, A212196/A181131, A291449/A291450, A290694/A290695, A291447/A291448.

Keyword

nonn,tabf,frac

Author

Peter Luschny, Aug 24 2017

Status

approved

A317624 Number of integer partitions of n where all parts are > 1 and whose LCM is n.

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 17, 1, 1, 1, 7, 1, 60, 1, 1, 1, 1, 1, 76, 1, 1, 1, 55, 1, 105, 1, 11, 10, 1, 1, 187, 1, 6, 1, 13, 1, 30, 1, 111, 1, 1, 1, 5043, 1, 1, 15, 1, 1, 230, 1, 17, 1, 242, 1, 4173, 1, 1, 12, 19, 1

Offset

0,13

Links

Antti Karttunen, Table of n, a(n) for n = 0..359

Index entries for sequences related to lcm's

Example

The a(20) = 5 partitions are (20), (10,4,4,2), (10,4,2,2,2), (5,5,4,4,2), (5,5,4,2,2,2).
The a(45) = 10 partitions:
  (45),
  (15,15,9,3,3), (15,9,9,9,3),
  (15,9,9,3,3,3,3), (15,9,5,5,5,3,3), (9,9,9,5,5,5,3),
  (15,9,3,3,3,3,3,3,3), (9,9,5,5,5,3,3,3,3), (9,5,5,5,5,5,5,3,3),
  (9,5,5,5,3,3,3,3,3,3,3).
From David A. Corneth, Sep 08 2018: (Start)
Let sum(t) denote the sum of elements of a tuple t. The tuples t with distinct divisors of 45 that have lcm(t) = 45 and sum(t) <= 45 are {(45) and (3, 9, 15), (3, 5, 9, 15), (3, 5, 9), (5, 9), (9, 15), (5, 9, 15)}. For each such tuple t, find the number of partitions of 45 - s(t) into distinct parts of t.
For the tuple (45), there is 1 partition of 45 - 45 = 0 into parts with 45. That is: {()}.
For the tuple (3, 9, 15), there are 4 partitions of 45 - (3 + 9 + 15) = 18 into parts with 3, 9 and 15. They are {(3, 15), (9, 9), (3, 3, 3, 9), (3, 3, 3, 3, 3, 3)}.
For the tuple (3, 5, 9), there are 4 partitions of 45 - (3 + 5 + 9) = 28 into parts with 3, 5 and 9; they are {(5, 5, 9, 9), (3, 3, 3, 5, 5, 9), (3, 5, 5, 5, 5, 5), (3, 3, 3, 3, 3, 3, 5, 5)}.
For the tuple (3, 5, 9, 15), there is 1 partition of 45 - (3 + 5 + 9 + 15) = 13 into parts with 3, 5, 9 and 15. That is (3, 5, 5).
The other tuples, (5, 9), (9, 15), and (5, 9, 15); they give no extra tuples. That's because there is no solution to the Diophantine equation for 5x + 9y = 45 - (5 + 9), corresponding to the tuple (5, 9) with nonnegative x, y.
That also excludes (9, 15); if there is a solution for that, there would also be a solution for (5, 9). This could whittle down the number of seeds even further. Similarly, (5, 9, 15) gives no solution.
Therefore a(45) = 1 + 4 + 4 + 1 = 10.
(End)
In general, there are A318670(n) (<= A069626(n)) such seed sets of divisors where to start extending the partition from. (See the second PARI program which uses subroutine toplevel_starting_sets.) - Antti Karttunen, Sep 08 2018

Mathematica

Table[Length[Select[IntegerPartitions[n], And[Min@@#>=2, LCM@@#==n]&]], {n, 30}]

Prog

(PARI)
strong_divisors_reversed(n) = vecsort(select(x -> (x>1), divisors(n)), , 4);
partitions_into_lcm(orgn, n, parts, from=1, m=1) = if(!n, (m==orgn), my(k = #parts, s=0); for(i=from, k, if(parts[i]<=n, s += partitions_into_lcm(orgn, n-parts[i], parts, i, lcm(m, parts[i])))); (s));
A317624(n) = if(n<=1, 0, partitions_into_lcm(n, n, strong_divisors_reversed(n))); \\ Antti Karttunen, Sep 07 2018
(PARI)
strong_divisors_reversed(n) = vecsort(select(x -> (x>1), divisors(n)), , 4);
partitions_into(n, parts, from=1) = if(!n, 1, if(#parts==from, (0==(n%parts[from])), my(s=0); for(i=from, #parts, if(parts[i]<=n, s += partitions_into(n-parts[i], parts, i))); (s)));
toplevel_starting_sets(orgn, n, parts, from=1, ss=List([])) = { my(k = #parts, s=0, newss); if(lcm(Vec(ss))==orgn, s += partitions_into(n, ss)); for(i=from, k, if(parts[i]<=n, newss = List(ss); listput(newss, parts[i]); s += toplevel_starting_sets(orgn, n-parts[i], parts, i+1, newss))); (s) };
A317624(n) = if(n<=1, 0, toplevel_starting_sets(n, n, strong_divisors_reversed(n))); \\ Antti Karttunen, Sep 08-10 2018

Crossrefs

Cf. A000837, A018818, A066874, A067538, A069626, A074761, A074971, A143773, A259936, A281116, A285572, A290103, A305566, A316429, A316431, A316432, A316433, A318670.

Keyword

nonn

Author

Gus Wiseman, Aug 01 2018

Status

approved

A326454 Irregular triangle read by rows: T(n,k) is the number of small Schröder paths such that the area between the path and the x-axis is equal to n and contains k down-triangles.

1, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 7, 5, 1, 9, 13, 1, 1, 11, 25, 8, 1, 13, 41, 28, 1, 1, 15, 61, 68, 11, 1, 17, 85, 136, 51, 1, 1, 19, 113, 240, 155, 15, 1, 21, 145, 388, 371, 86, 1, 1, 23, 181, 588, 763, 314, 19

Offset

0,7

Comments

A227543 is the companion triangle for Dyck paths.

Number of n triangle stacks, in the sense of A224704, containing k down- triangles.

A Schröder path is a lattice path in the plane starting and ending on the x-axis, never going below the x-axis, using the steps (1,1) rise, (1,-1) fall or (2,0) flat. A small Schröder path is a Schröder path with no flat steps on the x-axis.

The area between a small Schröder path and the x-axis may be decomposed into a stack of unit area triangles; the triangles come in two types: up-triangles with vertices at the lattice points (x, y), (x+1, y+1) and (x+2, y) and down-triangles with vertices at the lattice points (x, y), (x-1, y+1) and (x+1, y+1). See the illustration in the Links section for an example.

Links

Table of n, a(n) for n=0..56.

P. Bala, Illustration for row 5

P. Bala, The area beneath small Schröder paths: Notes on A224704, A326453 and A326454

Formula

O.g.f. as a continued fraction: A(q,d) = 1/(2 - (1 + q)/(2 - (1 + q^3*d)/(2 - (1 + q^5*d^2)/( (...) )))) = 1 + q + q^2 + q^3*(1 + d) + q^4*(1 + 3*d) + q^5*(1 + 5*d + d^2) + ... (q marks the area, d marks down-triangles).

Other continued fractions: A(q,d) = 1/(1 - q/(1 - q^2*d - q^3*d/(1 - q^4*d^2 - q^5*d^2/(1 - q^6*d^3 - (...) )))).

A(q,d) = 1/(1 - q/(1 - (q^2*d + q^3*d)/(1 - q^5*d^2/(1 - (q^4*d^2 + q^7*d^3)/(1 - q^9*d^4/(1 - (q^6*d^3 + q^11*d^5)/(1 - q^13*d^6/( (...) )))))))).

O.g.f. as a ratio of q-series: N(q,d)/D(q,d), where N(q,d) = Sum_{n >= 0} (-1)^n*d^(n^2)*q^(2*n^2 + n)/( (1 - d*q^2)*(1 - d^2*q^4)*...*(1 - d^n*q^(2*n)) )^2 and D(q,d) = Sum_{n >= 0} (-1)^n*d^(n^2 - n)*q^(2*n^2 - n)/( (1 - d*q^2)*(1 - d^2*q^4)*...*(1 - d^n*q^(2*n)) )^2.

Example

Triangle begins
  n\k|  0    1    2     3    4
------------------------------
   0 |  1
   1 |  1
   2 |  1
   3 |  1    1
   4 |  1    3
   5 |  1    5    1
   6 |  1    7    5
   7 |  1    9   13    1
   8 |  1   11   25    8
   9 |  1   13   41   28    1
  10 |  1   15   61   68   11
  ...

Crossrefs

Row sums A224704. Cf. A001003, A227543, A309086, A326453.

Keyword

nonn,tabf,easy

Author

Peter Bala, Jul 06 2019

Status

approved

A225174 Square array read by antidiagonals: T(m,n) = greatest common unitary divisor of m and n.

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 5, 1, 3, 1, 1

Offset

1,5

References

M. Lal, H. Wareham and R. Mifflin, Iterates of the bi-unitary totient function, Utilitas Math., 10 (1976), 347-350.

Links

Antti Karttunen, Table of n, a(n) for n = 1..20100; the first 200 antidiagonals of the array

Formula

T(m,n) = T(n,m) = A165430(n,m).

Example

Array begins
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, ...
1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, ...
1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, ...
1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, ...
1, 2, 3, 1, 1, 6, 1, 1, 1, 2, 1, 3, ...
1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, ...
...
The unitary divisors of 3 are 1 and 3, those of 6 are 1,2,3,6; so T(6,3) = T(3,6) = 3.

Maple

# returns the greatest common unitary divisor of m and n
f:=proc(m, n)
local i, ans;
ans:=1;
for i from 1 to min(m, n) do
if ((m mod i) = 0) and (igcd(i, m/i) = 1)  then
   if ((n mod i) = 0) and (igcd(i, n/i) = 1)  then ans:=i; fi;
fi;
od;
ans; end;

Mathematica

f[m_, n_] := Module[{i, ans=1}, For[i=1, i<=Min[m, n], i++, If[Mod[m, i]==0 && GCD[i, m/i]==1, If[Mod[n, i]==0 && GCD[i, n/i]==1, ans=i]]]; ans];
Table[f[m-n+1, n], {m, 1, 14}, {n, 1, m}] // Flatten (* Jean-François Alcover, Jun 19 2018, translated from Maple *)

Prog

(PARI)
up_to = 20100; \\ = binomial(200+1, 2)
A225174sq(m, n) = { my(a=min(m, n), b=max(m, n), md=0); fordiv(a, d, if(0==(b%d)&&1==gcd(d, a/d)&&1==gcd(d, b/d), md=d)); (md); };
A225174list(up_to) = { my(v = vector(up_to), i=0); for(a=1, oo, for(col=1, a, if(i++ > up_to, return(v)); v[i] = A225174sq((a-(col-1)), col))); (v); };
v225174 = A225174list(up_to);
A225174(n) = v225174[n]; \\ Antti Karttunen, Nov 28 2018

Crossrefs

See A034444, A077610 for unitary divisors of n.

Different from A059895.

Keyword

nonn,tabl,look

Author

N. J. A. Sloane, May 01 2013

Status

approved

A356167 Greatest common divisor of A003961(n) and the smallest positive k such that n divides k*A003961(k), where A003961 is fully multiplicative with a(p) = nextprime(p).

1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 7, 1, 1, 1, 1, 3, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 1, 1, 3, 11, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 5

Offset

1,10

Links

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

Index entries for sequences computed from indices in prime factorization

Formula

a(n) = gcd(A003961(n), A356164(n)).

Prog

(PARI)
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A356167(n) = for(k=1, oo, if((k*A003961(k))%n==0, return(gcd(A003961(n), k))));

Crossrefs

Cf. A003961, A191002, A356164, A356165, A356166.

Cf. also A345993.

Keyword

nonn,look

Author

Antti Karttunen, Jul 28 2022

Status

approved

A093421 Triangle read by rows: T(n,k) is the denominator of f(n, k) = (Product_{i = 0..k-1} (n-i))/(Sum_{i = 1..k} i) for 1 <= k <= n.

1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset

1,3

Links

Table of n, a(n) for n=1..105.

Formula

T(n,n) = denominator(f(n, n)) = denominator(2*(n-1)!/(n+1)).

Example

Triangle T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
  1;
  1, 3;
  1, 1, 1;
  1, 1, 1, 5;
  1, 3, 1, 1, 1;
  1, 1, 1, 1, 1, 7;
  1, 1, 1, 1, 1, 1, 1;
  1, 3, 1, 1, 1, 1, 1, 1;
  1, 1, 1, 5, 1, 1, 1, 1, 1;
  1, 1, 1, 1, 1, 1, 1, 1, 1, 11;
  ...

Crossrefs

Cf. A090585, A090586, A093415, A093420 (numerators), A093423.

Keyword

nonn,easy,frac,tabl,less

Author

Amarnath Murthy, Mar 30 2004

Extensions

Edited and extended by David Wasserman, Aug 29 2006

Status

approved

A143069 Least number k such that n*k has the fewest possible ones in its binary expansion.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 1, 27, 1, 1, 3, 3, 1, 41, 5, 19, 1, 565, 1, 1, 1, 1, 1, 1, 1, 7085, 27, 7, 1, 25, 1, 3, 3, 1, 3, 11, 1, 1, 41, 1, 5, 1266205, 19, 7, 1, 9, 565, 9099507, 1, 17602325, 1, 1, 1, 1, 1, 128207979, 1, 1, 1, 119, 1, 1, 7085, 1, 27, 5, 7, 13

Offset

1,11

Comments

a(n)=1 indicates that n is a sturdy number (A125121); that is, no multiple of n has fewer ones than the binary expansion of n. A086342(n) gives the least possible number of ones in the binary expansion of a multiple of n. Compare with A143073.

Links

T. D. Noe, Table of n, a(n) for n=1..1000

Trevor Clokie et al., Computational Aspects of Sturdy and Flimsy Numbers, arxiv preprint arXiv:2002.02731 [cs.DS], February 7 2020.

Keyword

nonn

Author

T. D. Noe, Jul 22 2008

Status

approved

A318449 Numerators of the sequence whose Dirichlet convolution with itself yields A001511, the 2-adic valuation of 2n.

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 35, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 1

Offset

1,9

Links

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

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A001511(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.

Mathematica

a1511[n_] := IntegerExponent[2n, 2];
f[1] = 1; f[n_] := f[n] = 1/2 (a1511[n] - Sum[f[d] f[n/d], {d, Divisors[ n][[2 ;; -2]]}]);
Table[f[n] // Numerator, {n, 1, 105}] (* Jean-François Alcover, Sep 13 2018 *)

Prog

(PARI)
up_to = 65537;
A001511(n) = 1+valuation(n, 2);
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&d<n, u[d]*u[n/d], 0)))/2); u}; \\ From A317937.
v318449_51 = DirSqrt(vector(up_to, n, A001511(n)));
A318449(n) = numerator(v318449_51[n]);

Crossrefs

Cf. A001511, A318450 (denominators).

Keyword

nonn,frac

Author

Antti Karttunen and Andrew Howroyd, Aug 29 2018

Status

approved